LIBRARY. 

OF    THE 

UNIVERSITY  OF  CALIFORNIA. 
Class 


"-••^•H   r'H-'B 

•  ^1 


A  "BROKEN"  TRANSIT. 
Length  of  axis,   34  inches.     Approximate  cost,   $1200. 


A   TEXT-BOOK 


OF 


FIELD   ASTRONOMY 

FOR  ENGINEERS. 


BY 

GEORGE  C.  COMSTOCK, 

Director  of  the  Washburn  Observatory, 
Professor  of  Astronomy  in  the   University  of  Wisconsin. 


FIRST    EDTTTON 
FIRST   THOUSAND. 


OF  THE 

UNIVERSITY 

OF 


NEW  YORK: 

JOHN  WILEY  &  SONS. 

LONDON:  CHAPMAN  &  HALL.  LIMITED. 
1903. 


Copyright,  190*, 

BY 

GEORGE  C.  COMSTOCK. 


ROBliRT    DRTTMMONP      PPTNTER,    NEW    YORK. 


PREFACE. 


THE  present  work  is  not  designed  for  professional 
students  of  astronomy,  but  for  another  and  larger  class 
found  in  technical  colleges.  For  many  years  it  has  been 
the  author's  duty  to  teach  to  students  of  engineering 
the  elements  of  practical  astronomy,  and  the  experience 
thus  acquired  has  gradually  produced  the  unconventional 
views  that  find  expression  in  the  present  text  and  which, 
to  the  author's  mind,  are  justified  by  the  following 
considerations : 

In  the  engineering  curriculum,  work  in  astronomy  is 
a  part  of  a  course  of  technical  and  professional  training 
of  students  who  have  no  purpose  to  become  astronomers. 
Under  these  circumstances  it  seems  the  duty  of  the 
instructor  to  select  for  presentation  those  parts  of 
astronomical  practice  most  closely  related  to  the  work 
of  the  future  engineer  and,  with  reference  to  the  narrow 
limits  of  time  allotted  the  subject,  to  keep  in  the  back- 
ground many  collateral  matters  that  are  of  primary  in- 
terest and  importance  to  the  student  of  astronomy  as  a 
science. 

The  parts  of  astronomical  practice  most  pertinent  to 

iii 


IV  PREFACE. 

engineering  instruction  seem  to  the  author  to  be  (a)  Train- 
ing in  the  art  of  numerical  computation ;  (6)  Training  in 
the  accurate  use  of  such  typical  instruments  of  precision 
as  the  sextant  and  the  theodolite,  with  special  refer- 
ence to  the  elimination  of  their  errors  from  the  results  of 
observation;  (c)  Determinations  of  time,  latitude,  and 
azimuth,  with  portable  instruments,  as  furnishing  sub- 
ject-matter through  which  a  and  b  may  be  conveniently 
realized.  If  this  work  is  to  be  done  during  the  single 
semester  usually  allowed  for  the  subject,  the  time  given 
to  its  theoretical  side,  spherical  astronomy,  must  be 
reduced  to  the  minimum  amount  compatible  with  the 
student's  intelligent  use  of  his  apparatus  and  formulas, 
and  in  the  present  work  this  pruning  of  the  theoretical  side 
has  been  carried  to  an  extent  that  would  be  unpardon- 
able in  the  training  of  an  astronomer,  but  which  appears 
necessary  and  proper  in  this  case. 

Since  many  engineering  students  acquire  from  the 
mathematical  curriculum  little  or  no  knowledge  of 
spherical  trigonometry  and  its  numerical  applications, 
the  first  chapter  of  the  work  is  devoted  to  a  brief  presen- 
tation of  the  elements  of  this  subject  with  special  refer- 
ence to  its  astronomical  uses  and  to  the  student's  acqui- 
sition of  good  habits  in  the  conduct  of  numerical  work. 
The  astronomical  problems  presented  in  the  following 
chapters  are  those  that  have  been  indicated  by  experience 
as  best  adapted  to  the  author's  own  pupils,  and  while 
many  of  the  methods  given  for  their  solution  are  not 
contained  in  the  current  text-books,  in  every  case  these 
are  either  methods  in  use  in  the  best  geodetic  surveys, 


PREFACE.  V 

or  such  as  have  been  repeatedly  tested  with  students  and 
found  well  suited  to  their  use.  These  methods  are  classi- 
fied in  the  text  as  rough,  approximate,  and  precise,  with 
respect  to  their  precision  and  the  corresponding  amount 
of  time  and  labor  required  for  their  application,  and  the 
student  is  advised  not  to  use  the  refined  and  laborious 
methods  when  only  a  rough  result  is  required. 

As  a  rule,  in  the  development  of  formulae  no  attempt 
has  been  made  to  deal  with  the  general  case  when  the 
solution  of  a  particular  case  would  suffice  for  the  prob- 
lem in  hand;  e.g.,  the  earth's  compression  is  ignored 
in  treating  of  the  effect  of  parallax,  since  its  influence 
is  vanishingly  small  in  the  great  majority  of  cases  that 
the  student  will  ever  encounter,  and  cases  in  which  this 
influence  is  of  sensible  amount  should  be  avoided  by  the 
instructor.  A  more  serious  omission,  but  one  required 
by  the  general  plan  of  the  work,  is  found  in  the  theory 
of  the  transit  instrument,  Chapter  IX,  where  broken 
transits,  thread  intervals,  curvature  of  a  star's  apparent 
path,  flexure,  etc.,  are  passed  by  without  treatment  or 
even  suggestion.  They  are  not  required  for  the  begin- 
nings of  work  with  a  transit  instrument,  and  therefore 
constitute  a  part  of  more  advanced  study  than  is  here 
contemplated.  As  a  partial  guide  to  such  study  there 
is  given  upon  a  subsequent  page  a  list  of  references  to 
works  that  may  be  consulted  with  profit  by  the  student 
who  seeks  a  more  complete  knowledge  of  the  processes 
of  practical  astronomy. 

The  adopted  notation  follows,  with  only  slight  devia- 
tions, that  of  Chauvenet,  to  whose  elaborate  treatise 


vi  PREFACE. 

upon  Spherical  and  Practical  Astronomy  the  author  is 
under  obligations  that  are  common  to  every  present-day 
writer  upon  those  subjects.  His  thanks  are  also  due 
to  many  of  his  former  pupils,  and  in  particular  to  Dr. 
S.  D.  Townley  and  Dr.  Joel  Stebbins,  who  have  read 
and  criticised  portions  of  his  manuscript. 


TABLE  OF  SYMBOLS. 


THE  following  table  contains  a  brief  explanation  of  the 
principal  symbols  employed  in  the  text,  with  references 
to  the  page  at  which  they  are  respectively  defined.  There 
are  omitted  from  the  table  a  considerable  number  of 
symbols  employed  only  in  immediate  connection  with 
their  definition. 

Mathematical. 

2  p.  154  Summation  symbol. 

[  ]  ii  The  enclosed  number  is  a  logarithm. 

Coordinates,  etc. 

h  p.    26  Altitude. 

0  25  Zenith  distance.     Complement  of  h. 

A  26  Azimuth,  reckoned  from  south. 

AN  65  Azimuth,  reckoned  from  north. 

t  26  Hour  angle. 

d  26  Declination, 

a  26  Right  ascension. 

$  29  Latitude. 

A  42  Longitude. 

5  52  Sun's  semi-diameter. 
P  53  Horizontal  parallax. 

Time. 

6  p.    30     Sidereal  time. 
M  41     Mean  solar  time. 

T  44  Time  shown  by  a  chronometer,  whether  right  or  wrong. 

AT  44  Chronometer  correction. 

p  45  Chronometer  rate. 

E  40  Equation  of  time. 

D  41  Date  of  an  observation. 

V  40  Date  of  conjunction,  mean  sun  with  vernal  equinox. 


vin  TABLE  OF  SYMBOLS. 

Q  p.    42     Sidereal  time  of  mean  noon,  as  given  in  almanac. 

Ql  42     Sidereal  time  of  mean  noon  reduced  to  the  local  merid- 

ian. 

a",  a'"          41     Daily  gain  of  sidereal  upon  mean  solar  time. 
Rough  and  Approximate  Determinations. 

a  p.    70     Tabular  difference  of  azimuth,  Polaris  and  north  pole. 

b  70     Tabular  difference  of  altitude,  Polaris  and  north  pole. 

Flt  F2  72     Factors  to  transform  a  and  b  into  their  true  local  values. 

A'  71     Difference  of  right  ascension,  Polaris  and  mean  sun. 

E  71     Date  of  conjunction,  Polaris  and  mean  sun. 

M  91     An  approximate  value  of  the  chronometer  correction, 

AT. 

m  91     An  approximate  index  correction  of   the  horizontal 

circle. 

x,  y  91     Corrections  to  transform  u  and  m  into  true  values. 

g  92     A  coefficient,  equals  —T-. 

D  93     An  auxiliary  used  in  the  computation  of  g. 

T  91     An  hour  angle  reckoned  toward  the  east. 

Instruments. 

R,  r        p.  118     Circle  or  micrometer  readings. 

d  100     Value  of  half  a  level  division. 

k  159     Value  of  one  revolution  of  a  micrometer. 

f  118     Deviation  of  a  vertical  axis  from  the  true  vertical. 

b',  b"  114     Components  of  f  parallel  and  perpendicular  to  hori- 

zontal axis. 

i  118     Deviation,  from  90°,  of  angle  between  axes  of  theodo- 

lite. 

w  120     Complement  of  spherical  angle  at  zenith  between  hori- 

zontal axis  and  line  of  sight. 

a  171     Deviation  of  horizontal  axis  from  true  east  and  west. 

b  171     Deviation  of  horizontal  axis  from  true  level. 

c  171     Collimation  constant. 

A,  B  tC       173     Mayer's  transit  factors. 

Cm  162     Calibration  correction  to  micrometer-screw. 

Precise  Determinations. 

L  p.  144     Correction  to  equal  altitudes.      (Time.) 

ARQ  148     Correction  to  equal  altitudes.      (Azimuth.) 

g,  k  149     Auxiliaries  used  in  computation  of  azimuths. 

s  1 60     Auxiliary  used  in  computation  of  differential  refrac- 

tion. 

/  83     Auxiliary  used  in  computation  of  reduction  to  merid- 

ian. 

o  189     Auxiliary  collimation  coefficient. 


TABLE  OF  CONTENTS. 


CHAPTER  I. 

PAGE 

INTRODUCTORY i 

Spherical  trigonometry,.    Approximate  formulae.    Numeri- 
cal computations.     Logarithmic  tables.     Limits  of  accuracy. 


CHAPTER  II. 
COORDINATES 22 

Fundamental  concepts.  Definitions.  Notation.  Table  of 
coordinates.  Transformation  of  coordinates. 

CHAPTER  III. 
TIME 35 

Three  time  systems.  Longitude.  Conversion  of  time. 
Chronometer  corrections.  The  almanac. 

CHAPTER  IV. 

CORRECTIONS  TO  COORDINATES 49 

Dip  of  horizon.  Refraction.  Semi-diameter.  Parallax. 
Diurnal  aberration. 

CHAPTER  V. 
ROUGH  DETERMINATIONS 59 

Latitude  from  meridian  altitude.  Time  and  azimuth  from 
single  altitude.  Meridian  transits  for  time.  Orientation  and 
latitude  by  Polaris. 

ix 


TABLE  OF  CONTENTS. 


CHAPTER  VI. 

PAGE 

APPROXIMATE  DETERMINATIONS 79 

Circum-meridian  altitudes  for  latitude.  Time  from  single 
altitude.  Azimuth  observations  at  elongation.  Time  and 
azimuth  from  two  stars. 

CHAPTER  VII. 
INSTRUMENTS 99 

The  spirit-level.  Value  of  half  a  level  division.  Theory  of 
the  theodolite.  Repetition  of  angles.  The  sextant.  Chro- 
nometers. 

CHAPTER  VIII. 

ACCURATE  DETERMINATIONS 141 

Time  by  equal  altitudes.  Precise  azimuth  with  theodolite. 
Zenith-telescope  latitudes. 

CHAPTER  IX. 

THE  TRANSIT  INSTRUMENT 168 

Preliminary  adjustments.  Theory  of  the  transit.  Ordinary 
method  for  time  determinations.  Personal  equation.  Methods 
and  accuracy  of  observation.  Time  determination  with 
reversal  on  each  star.  Azimuth  of  terrestrial  mark. 

BIBLIOGRAPHY 196 

ORIENTATION  TABLES 197 


FIELD  ASTRONOMY. 

CHAPTER  I. 

INTRODUCTORY. 

i.  Spherical  Trigonometry. — Any  three  points  on  the 
surface  of  a  sphere  determine  a  spherical  triangle,  whose 
sides  are  the  arcs  of  great  circles  joining  these  points, 
and  whose  angles  are  the  spherical  angles  included  be- 
tween these  arcs;  e.g.,  on  the  surface  of  the  earth, 
assumed  to  be  spherical  in  shape,  the  north  pole,  the  city 
of  St.  Louis,  and  the  borough  of  Greenwich,  England,  are 
three  points  making  a  spherical  triangle,  two  of  t  whose 
sides  are  the  arcs  of  meridians  joining  St.  Louis  and  Green- 
wich to  the  pole ;  the  third  side  being  the  arc  of  a  great 
circle  connecting  St.  Louis  and  Greenwich,  and  measur- 
ing by  its  length  the  distance  of  one  place  from  the  other. 
The  spherical  angle  at  the  pole  between  the  two  meridians 
is  the  longitude  of  St.  Louis,  while  the  angle  at  St.  Louis 
between  its  meridian  and  the  third  side  of  the  triangle 
represents  the  direction  of  Greenwich  from  St.  Louis,  a 
certain  number  of  degrees  east  of  north.  The  particular 
number  of  degrees  in  this  angle  is  to  be  found  by  solving 


2  FIELD  ASTRONOMY. 

the  triangle,  i.e.,  determining  the  magnitude  of  its  un- 
known parts  by  means  of  the  known  parts,  and  in  this 
case  we  may  suppose  these  known  parts  to  be  the  differ- 
ence of  longitude  between  the  two  places,  and  the  distance 
of  each  place  from  the  north  pole,  i.e.,  the  complement 
of  its  latitude. 

The  formulae  required  for  the  solution  of  a  spherical 
triangle  are  best  derived  by  the  methods  of  analytical 
geometry,  and  in  Fig.  i  we  assume  a  spherical  triangle, 


FIG.  i. 

ABC,  situated  on  the  surface  of  a  sphere  whose  centre  is 
at  0,  and  we  adopt  0  as  the  origin  of  a  system  of  rect- 
angular coordinates,  in  which  the  axis  OX  passes  through 
the  vertex,  A,  of  the  triangle,  OY  lies  in  the  plane  A  OB, 
and  OZ  is  perpendicular  to  that  plane.  From  the  vertex 
C  let  fall  upon  the  plane  OAB  the  perpendicular  CP,  and 
from  P  draw  PS  perpendicular  to  OX  and  join  the  points 
C,  S,  thus  obtaining  the  right-angled  plane  triangle  CPS. 


INTRODUCTORY.  3 

The  lines  05,  SP,  PC  are  respectively  the  x,  y,  and  z 
coordinates  of  the  point  C,  and  OC,  which  we  shall  repre- 
sent by  the  symbol  r,  is  the  radius  of  the  sphere. 

It  is  evident  from  the  construction  that  the  points 
0,  5,  A,  and  C  all  lie  in  the  same  plane.  Also,  0,  5,  A, 
B,  and  P  lie  in  another  plane,  and  the  angle  between  these 
two  planes  is  measured  both  by  the  spherical  angle  BAG 
and  by  the  plane  angle  CSP,  and  these  angles  must  there- 
fore be  equal  each  to  the  other.  We  may  now  express 
the  coordinates  of  the  point  C  in  terms  of  the  sides,  a,  6,  c, 
and  angles,  A,  B,  C,  of  the  spherical  triangle  as  follows: 

OS=x  =  r  cos  6, 

SP=y=r  sin  b  cos  A,  (i) 

PC  =  z  =  r  sin  b  sin  A. 

If  the  axis  of  x,  instead  of  passing  through  A,  had  been 
made  to  pass  through  B,  as  is  shown  by  the  broken  line 
OX' ,  the  axis  of  Z  remaining  unchanged,  we  should  have 
had  for  the  coordinates  of  C  in  this  system, 

yf  =     r  cos  a, 

y  =  —  r  sin  a  cos  B,  (2) 

^  =     r  sin  a  sin  B. 

For  the  sake  of  simplicity  each  angle  of  the  triangle  ABC 
has  been  made  less  than  90°,  and  the  point  P,  therefore, 
falls  between  the  axes  OX,  OX' ,  thus  giving  y  and  y' 
opposite  signs,  as  shown  above. 

It  is  evident  from  the  figure  that  the  relations  between 
#>  #',  y>  y'i  are  those  furnished  by  the  formulae  for  the 


4  FIELD  ASTRONOMY. 

transformation  of  coordinates  in  a  plane,  when  the  origin 
remains  unchanged  and  the  axes  are  revolved  through  an 
angle,  which  in  this  case  is  measured  by  the  side  c  of  the 
spherical  triangle.  We  have,  therefore, 

z'=z, 

y'  =y  cos  c  —  x  sin  c,  (3) 

x'  =  y  sin  c  +  x  cos  c ; 

and  introducing  into  these  equations  the  values  of  the 
coordinates  above  determined  and  dividing  through  by 
r,  we  obtain  the  following  relations  among  the  sides  and 
angles  of  the  triangle : 

sin  a  sin  B  =  sin  b  sin  A, 

sin  a  cos  B  =  cos  b  sin  c  —  sin  b  cos  c  cos  A,  (4) 

cos  a  =  cos  b  cos  c  +  sin  b  sin  c  cos  A . 

These  are  the  fundamental  equations  of  spherical 
trigonometry  and  hold  true  not  only  for  the  particular 
triangle  for  which  they  have  been  derived,  but  for  every 
spherical  triangle,  whatever  its  shape  or  size. 

2.  Numerical  Applications  of  Equations  4. — We  proceed 
to  apply  these  equations  to  the  logarithmic  solution  of 
the  triangle  above  described,  premising  that  in  this  solu- 
tion the  signs  of  all  the  trigonometric  functions  must  be 
carefully  heeded,  since  upon  them  depend  the  quadrants, 
first,  second,  third,  or  fourth,  in  which  the  unknown  parts 
of  the  triangle  are  to  be  found.  In  this  connection  we 
shall  reserve  the  signs  +  and  —  for  natural  numbers  and 
place  after  a  logarithm  the  letter  n  whenever  the  number 
corresponding  to  the  logarithm  is  negative.  The  student 


INTRODUCTORY. 


should  accustom  himself  to  this  practice,  since  it  is  the 
one  in  general  use. 

The  assumed  data  of  the  problem  are  : 


Angular  distance,  Greenwich  to  Pole.  ...  b  =3  8°.  5, 
Angular  distance,  St.  Louis  to  Pole  .....  c=5i°.4r 
Spherical  angle  at  North  Pole  ..........  A  =90°.  4. 

and  these  data  we  treat  as  follows  : 


Logarithms. 
sin  A  =0.000 
(ssi-n  6=9.794 
cos  A  =7.844?* 
cos  (7=9.795 
sin  6cos_.A  =7.638^ 

cos  6  =  9.894 
^sin  £=9.893 


SOLUTION. 

Numbers. 

cos  b  sin  c    =+0.613 
sin  b  cos  c  cos  A  =  —0.003 

cos  b  cos  c  =  +0.489 
sin  b  sin  c  cos  A  =  —0.003 


Logarithms, 
sin  a  sin  .6=9.794 

9-854 
sm  a  cos±?  =9.785 

-    #=45°-6 
sin  a  =9.940 

a  =60°.  9* 


log  cos  a  =     9.686 

In  the  solution  printed  above,  the  student  should 
examine  the  orderly  manner  of  the  arrangement.  Each 
number  is  labelled  to  show  what  it  is,  and  from  these 
labels  we  see  that  the  first  column  contains  the  logarithms 
of  the  several  trigonometric  functions  that  appear  in  the 
second  members  of  Equations  4.  The  second  column 
contains  natural  numbers  representing  the  values  of  the 
several  terms  contained  in  these  second  members.  These 
are  obtained  by  adding  the  proper  logarithms  shown  in 
the  first  column,  and  looking  out  the  corresponding  num- 
bers in  the  tables.  An  expert  computer  will  do  this  work 
"in  his  head"  without  writing  down  a  figure  that  is  not 
shown  in  the  printed  solution. 

At  the  bottom  of  the  second  column  is  given  log  cos  a, 
obtained  by  looking  out  the  logarithm  of  the  sum  of  the 
two  numbers  that  stand  just  above  it.  This  sum  being 
positive  shows  that  the  side  a  lies  in  either  the  first  or 


6  FIELD  ASTRONOMY. 

fourth  quadrant,  but  it  alone  cannot  decide  between  these 
two  possibilities.  We  must  now  have  recourse  to  the 
third  column,  which  gives  the  logarithms  of  the  products, 
sin  a  sin  B  and  sin  a  cos  B,  as  derived  from  the  first  and 
second  columns,  and  indicates  that  these  products  are 
positive  quantities,  since  no  n  is  appended  to  either  of  the 
logarithms.  The  products  being  positive,  the  factors 
sin  a,  sin  B,  and  cos  B  must  all  have  like  signs,  and  assum- 
ing, temporarily,  that  sin  a  is  a  positive  quantity  we 
find  that  B  must  lie  in  the  first  quadrant,  since  sin  B  and 
cos  B  are  positive"  numbers.  To  obtain  its  numerical 
value  we  divide  sin  a  sin  B  by  sin  a  cos  B  (subtract 
mentally  the  corresponding  logarithms)  and  find,  as  the 
result  of  the  division,  log  tan  B  =  0.009.  This  furnishes 
the  value  of  B  given  in  the  solution,  and  fixes  as  the 
direction  Of  Greenwich  from  St.  Louis,  N.  4 5°. 6  E. 

Now,  looking  up  in  the  logarithmic  tables  the  value 
of  sin  B  (log  sin  B  =  9. 8 5 4),  and  dividing  it  out  from 
sin  a  sin  B,  we  obtain  the  value,  9.940,  given  in  the  solu- 
tion for  sin  a.  This  number  might  equally  have  been 
obtained  by  looking  up  in  the  tables  the  value  of  cos  B 
( =  9.845)  and  dividing  it  out  from  sin  a  cos  B,  and  with 
reference  to  this  double  possibility  the  label  for  the  line 
between  sin  a  sin  B  and  sin  a  cos  B  is  omitted,  it  being 
understood  that  either  sin  B  or  cos  B,  whichever  is  the 
greater  of  the  two,  will  be  entered  here  and  used  in  the 
proper  manner  to  obtain  sin  a.  The  value  of  log  cos  a 
is  given  in  the  middle  column,  and  both  sin  a  and  cos  a 
being  positive  numbers,  a  is  to  be  taken  in  the  first  quad- 
rant. The  agreement  between  the  numerical  values  of  a 


INTRODUCTORY.  7 

furnished  by  the  sine  and  cosine,  is  a  check  upon  the  accu- 
racy of  the  computation,  and  an  asterisk  or  check-mark 
is  placed  after  the  value  of  a  to  show  that  this  check  has 
been  applied  and  found  satisfactory. 

In  determining  the  quadrant  of  B,  sin  a  was  assumed 
to  be  a  positive  number.  It  might  equally  well  have  been 
assumed  a  negative  number,  which  would  have  made 
sin  B  and  cos  B  both  negative,  and  would  have  furnished 
as  the  solution  of  the  triangle  B  =  225°. 6,  a  =  299°.i. 
This  is  also  a  correct  result,  for  if  we  travel  from  St.  Louis 
in  the  direction  N.  2  2  5°. 6  E.,  over  an 'arc  of  a  great  circle 
299°.  i  long,  we  shall  find  Greenwich  at  the  end  of  it. 
The  first  solution  represents  the  least  distance,  the  second 
solution  the  greatest  distance,  on  the  surface  of  the  sphere, 
between  the  two  points,  and  as  a  matter  of  convenience 
it  is  customary  to  use  the  first  solution  and  to  assume  that 
sin  a  is  a  positive  number. 

3.  Analytical  Applications  of  Equations  4. — Equations 
4  suffice  for  the  solution  of  any  triangle  in  which  there 
are  given  two  sides  and  the  included  angle,  but  they  are 
not  immediately  applicable  when  other  parts  of  the  tri- 
angle are  the  data  of  the  problem,  e.g.,  when  the  three 
sides  are  given  and  the  angles  are  required.  A  large  part 
of  spherical  trigonometry,  therefore,  consists  in  purely 
analytical  transformations  of  these  equations  into  forms 
adapted  to  different  data.  For  the  particular  case  above 
suggested,  a,  6',  and  c  given,  we  find  from  the  last  of 
Equations  4 

cos  a  —  cos  b  cos  c 

cosA .__.___  (S) 


FIELD  ASTRONOMY. 

by  means  of  which  the  angle  A  may  be  computed  with 
the  given  data,  and  similar  equations  may  be  written  for 
the  other  angles. 

By  transformations  more  tedious  than  difficult,  and 
involving  the  introduction  of  two  auxiliary  quantities, 
denned  below  by  Equations  6,  we  may  change  Equation  5 
into  a  form  more  convenient  for  computation  when  all 
three  of  the  angles  are  to  be  determined  (see  any  treatise 
on  spherical  trigonometry  for  the  analytical  processes 
involved) .  As  a  result  of  these  transformations  we  have 
the  following  auxiliaries  and  the  solution  involving  them : 


: 1 —, (6) 

n.(s—  a)  sin (5—  b)  sin($-<c) 
cot  %A=k  sin  (s—  a),  (7) 

with  corresponding  expressions  for  the  other  angles, 

cot  %B  =  k  sin  (s  —  b), 
cot  \C  =  k  sin  {s  —  c) . 

Right-angled  Spherical  Triangles. — Since  Equations  4 
hold  true  for  all  spherical  triangles,  we  may  apply  them 
to  the  special  case  of  a  triangle  right-angled  at  A,  i.e.,  one 
in  which  the  angle  A  equals  90°.  We  shall  then  have 
sin  A=i,  cos  A  =o,  and  with  these  special  values  we 
obtain  by  substitution  the  following  equations,  which 
should  be  compared  with  the  corresponding  formulae 
of  plane  trigonometry : 


INTRODUCTORY. 


From  the  first  equation,  sin  B  —  —r 


sm  a 


From  first  and  second  equations,  tan  B  =  an    . 

sm  c      (8) 

From  second  and  third  equations,  cos  B  =  ~  -  . 

tana 

From  the  third  equation,        cos  a  =  cos  b  cos  c. 

These  equations  together  with  those  derived  in  the 
preceding  sections,  while  far  from  covering  the  whole  field 
of  spherical  trigonometry,  will  be  found  sufficient  for  the 
purposes  of  this  work. 

4.  Approximate  Formulae.  —  In  an  important  class  of 
cases  all  the  preceding  formulae  may  be  greatly  simplified 
for  numerical  use  as  follows:  It  is  shown,  in  treatises  on 
the  differential  calculus,  that  the  trigonometric  functions 
may  be  developed  in  series;  e.g., 

x3      x5 
sin  x=x  —  -7-H  ----  etc., 

O         I2O 

X3       2X5 

tan*  =  #  +  —  +  —  +  etc.,  (9) 

O  3 

x2      x* 


2  24 

where  x  is  expressed  in  radians  (one  radian  =  5  7°.3, 
=  3437'-75>  =2o6264".8).  When  the  angle  x  does  not 
exceed  a  few  minutes  of  arc,  x  radians  is  a  small  fraction, 
and  its  powers,  x2,  x3,  etc.,  are  still  smaller  quantities,  so 
that  in  these  series  we  may  suppress  all  terms  save  the 
first,  or  all  terms  save  the  first  two,  and  the  error  pro- 
duced by  neglecting  these  terms  of  a  higher  order^as 


10  FIELD  ASTRONOMY. 

they  are  called,  is  approximately  measured  by  the  first 
term  thus  neglected.  For  illustration  we  assume  x  =  i°, 
and  turning  this  into  radians  find  the  results  shown  in 
the  following  short  table  : 

Radians.  Arc. 


§x2  =0.0001523  +  =  3i".42  XX 

%x3  =  o.  0000009  +  =     o".i8 
s\:£4  =0.0000000  -f-  =     o".oo 

It  appears  from  the  value  of  J#3,  here  given,  that  if  we 
are  prepared  to  tolerate  in  our  work  an  error  of  one  part 
in  a  million  we  may,  for  an  arc  of  i°,  substitute  the  arc 
itself  in  place  of  its  sine,  in  any  formula  where  the  latter 
occurs  ;  and  similarly  (from  the  value  of  J#2)  we  may  sub- 
stitute unity  in  place  of  the  cosine  of  an  arc  of  i°,  if  we  are 
willing  to  admit  an  error  of  one  part  in  seven  thousand. 
Expressed  in  arc  these  errors  are  as  shown  in  the  table, 
o".2  and  3  1  ".4  respectively,  and  with  reference  to  these 
numbers  we  may  establish  the  approximate  relations: 
the  square  of  a  degree  equals  a  minute;  the  cube  of  a 
degree  equals  a  second  ;  and  find  readily,  from  these  rela- 
tions, the  square  and  cube  of  any  small  arc,  and  thus 
decide  whether,  in  a  given  case,  these  quantities  may  or 
may  not  be  neglected.  For  example:  if  x  =  2°,  we  find 
#2  =  4/,  #3  =  8",  and  for  any  work  in  which  the  data  can 
be  depended  upon  to  the  nearest  minute  only,  we  may 
assume  sin  x—xt  but  we  cannot  assume  cos  x  =  i  without 
sacrificing  some  of  the  accuracy  contained  in  the  data. 

It  must  be  constantly  borne  in  mind  that  the  series 
given  above  are  expressed  in  radians,  and  that  when 
applied  numerically,  x  and  its  powers  must  be  trans- 


INTRODUCTORY.  11 

formed  from  arc  into  radians  by  dividing  by  the  appro- 
priate factors  given  above;  e.g., 

x" 

x  (radians)  =  — -— — -  ( 1 1 ) 

206264.8 

The  divisor  given  above  is  numerically  equal  to  the 
reciprocal  of  the  sine  of  i",  and  in  place  of  the  preceding 
equation  it  is  customary  to  write 

x  (radians)  =x"  sin  i".  (12) 

As  these  numerical  factors  are  of  frequent  use,  we  record 
here  their  values : 

log  sin  i"  =  4 . 6855 749  -  10, 
206264. 8  =  [5. 3144251]. 

Observe  the  peculiar  notation  of  the  last  line.  The 
brackets  indicate  that  the  number  placed  within  them  is 
a  logarithm,  and  the  equation  asserts  that  this  bracketed 
number  is  the  logarithm  of  the  first  member  of  the  equa- 
tion. This  use  of  the  brackets  is  very  common  and 
should  be  remembered. 

We  may  apply  to  the  equations  of  spherical  trigo- 
nometry the  principles  here  developed,  and  assuming  that 
the  sides,  a,  6,  c,  do  not  much  exceed  i°,  i.e.,  that  for  a 
triangle  on  the  surface  of  the  earth  the  vertices  of  the 
angles  are  not  more  than  sixty  or  seventy  miles  apart,  we 
shall  find  that  Equations  8  become 

sin  B  —  —t     cos  J3  =  -,     tan  B  =  -,     a2  =  b2  +  c2. 
a  Q>  c 


12  FIELD  ASTRONOMY. 

These  are  the  formulae  of  plane  trigonometry,  and  indicate 
that  small  spherical  triangles  may  be  treated  as  if  they 
were  plane. 

The  use  of  these  approximate  relations  is  not  limited 
to  the  solution  of  triangles,  but  they  may  be  applied  to 
the  trigonometric  functions  of  any  small  angle  wherever 
found,  and  we  shall  have  frequent  occasion  to  use  them 
in  the  following  pages. 

5.  Numerical  Computations. — Engineer  and  astronomer 
alike  should  acquire  the  art  of  rapid  and  correct  compu- 
tation, and  as  a  means  to  that  end  there  will  be  found 
on  subsequent  pages  examples  of  numerical  work  which 
should  be  studied  with  reference  to  their  arrangement  and 
the  order  in  which  the  several  processes  were  executed. 
Often  the  order  in  which  this  work  was  done  is  not  the 
order  in  which  the  numbers  appear  upon  the  printed 
page,  although  their  arrangement  upon  the  page  always 
follows  exactly  the  original  computation,  and  in  no  case 
is  to  be  regarded  as  a  mere  summary  of  results,  picked 
out  and  rearranged  after  the  actual  ciphering  had  been 
performed.  For  illustration  we  revert  to  the  example 
of  §  2,  and  note  that  sin  A  is  the  first  number  written  in 
the  solution  and  sin  b  stands  second.  But  the  second 
number  actually  written  down  in  the  computation  was 
cos  A,  instead  of  sin  b,  for,  having  found  the  place  in 
which  to  look  up  sin  A,  it  is  more  convenient  and  more 
economical  to  look  up  cos  A  at  once,  while  the  tables  are 
open  at  the  right  place,  rather  than  to  turn  away  for 
something  else  and  then  have  again  to  find  the  page 
and  place  corresponding  to  the  angle  A.  Having  finished 


INTRODUCTORY.  13 

with  the  required  functions  of  A,  sin  b  was  next  looked 
out  and  was  followed  by  cos  6,  although  this  required 
the  computer  to  skip  two  intervening  lines  of  the  com- 
putation and,  temporarily,  to  leave  them  blank. 

The  general  principle  here  observed  is :  When  a  table 
is  open  at  a  given  place,  look  up,  before  leaving  it,  all 
that  is  to  be  taken  from  that  place.  In  order  to  do  this 
it  is  necessary  to  block  out  the  computation  in  advance, 
and  this  was  done  in  the  case  under  consideration,  every 
label,  from  the  initial  sin  A  of  the  first  column  to  the 
concluding  a  of  the  last  column,  being  written  in  its 
appropriate  place  before  a  number  was  set  down  or  the 
logarithmic  table  opened.  The  form  of  computation 
thus  prearranged  is  called  a  schedule,  and  it  is  to  be 
strongly  urged  upon  the  student  as  a  measure  of  econ- 
omy and  good  practice,  that  he  should  draft,  at  the 
beginning  of  each  computation,  a  complete  schedule,  in 
which  every  number  to  be  employed  shall  be  assigned 
the  place  most  convenient  for  its  use.  In  general  the 
beginner  will  not  be  able  to  do  this  without  assistance 
from  an  instructor,  or  from  models  suitably  chosen,  and 
for  the  purposes  of  the  present  work  the  numerous  exam- 
ples contained  in  the  text  may  be  taken  as  such  models. 

Some  cardinal  points  in  the  arrangement  of  a  good 
schedule  are  as  follows: 

(A)  Make  it  short  but  complete.  Do  as  much  of  the 
work  "in  your  head"  as  can  be  done  without  unduly 
burdening  the  mind,  and  write  upon  paper  only  the 
things  that  are  necessary.  But  all  things  that  are  to  be 
written  should  have  places  assigned  thejn  in  the  schedule. 


14  FIELD  ASTRONOMY. 

No  side  computations,  upon  another  piece  of  paper, 
should  be  allowed,  and  the  entire  work  should  be  so 
arranged  and  labelled .  that  a  stranger  can  follow  it  and 
tell  what  has  been  done. 

(B)  When  the  same  quantity  is  to  be  used  several 
times  in  a  computation  (sin  b  appears  as  a  factor  in  three 
different  terms  of  the  preceding  example)  the  schedule 
should  be  so  arranged  that  the  number  need  be  written 
only  once,  e.g.,  since  sin  b  is  to  be  multiplied  by  both 
sin  A  and  cos  A   it  is  placed  between   these   numbers 
in  the  schedule,  and  for  a  similar  reason  the  product 
sin  b  cos  A  is  placed  between  cos  c  and  sin  c.     In  adding 
the    logarithms   to   form   the   product   sin  b  sin  c  cos  A , 
cover  cos  b  with  a  pencil  or  penholder  and  the  addition 
will  be  as  easily  made  as  if  the  intervening  number  were 
not  present. 

(C)  Frequently,  several  similar  computations  are  to 
be  made  with  slightly  different  data,  e.g.,  it  may  be  re- 
quired to  find  the  direction  and  distance  of  half  a  dozen 
American    cities    from    Greenwich.     A    single    schedule 
should  then  be  prepared  and  the  several  computations 
should  be  carried  on  simultaneously,  in  parallel  columns, 
all  placed  opposite  the  same  schedule;  e.g.,   look  out 
sin  A  and  cos  A  for  all  six  places  before  proceeding  to 
find  sin  b  for  any  of  them,  etc.     In  this  particular  case 
sin  b  and  cos  b,  depending  on  the  latitude  of  Greenwich, 
are  the  same  for  all  the  solutions,  and  instead  of  writing 
their  values  in  each  column,  they  should  be  written  upon 
the  edge  of  a  slip  of  paper  and  moved  along  from  column 
to  column  as  needed.     As  a  memorandum  for  future 


INTRODUCTORY.  15 

reference  they  should  also  be  written  in  one  column  of 
the  computation.  Practise  this  device  whenever  the 
same  number  is  to  be  used  in  several  different  places. 
See  §§36  and  40  for  examples  of  two  computations  de- 
pending upon  a  single  schedule. 

6.  The  Trigonometric  Functions. — There  is  opened  to 
the  inexperienced  computer  an  abundant  opportunity  for 
error  in  looking  out  from  the  tables  the  trigonometric 
functions  of  angles  not  lying  in  the  first  quadrant.  The 
best  mode  of  guarding  against  such  errors  is  the  acqui- 
sition of  fixed  habits  of  procedure,  so  that  the  same  thing 
shall  always  be  done  in  the  same  way,  and  to  this  end 
the  following  simple  rules  may  be  adopted : 

(1)  For  any  odd-numbered  quadrant,  first,  third,  etc. 
Reduce  the  given  angle  to  the  first  quadrant  by  casting 
out  the  nines  from  its  tens  and  hundreds  of  degrees  (add 
these  digits  together  and  repeat  the  addition  until  the 
sum  is  reduced  to  a  single  digit,  less  than  nine),  and  look 
up  the  required  function  of  the  reduced  arc. 

(2)  For  any  even-numbered  quadrant,  second,  fourth, 
etc.     Reduce  the  angle  to  the  first  quadrant,  as  above, 
and  look  out  the  function  complementary  to  the  one 
given.     The  algebraic  sign  of  the  function  is,  of  course, 
in  all  cases  determined  by  the  quadrant  in  which  the 
original  angle  falls. 

See  the  following  applications  of  these  rules: 

Quadrant.  Required.                      Equivalent.  Process, 

ad,    Even  005144°  29'  =  —sin  54°  29  1+4  =5 

3d,    Odd  tan  264°  33'  =  +  tan  84°  33'  2+6=8 

4th, Even  sin  316°  51'  =  -00546°  51'  3  +  I==4 

Sth,  Odd  cot  414°  i »'  =  + cot  54°  18'  4  +  i=5 

6th,  Even  tan  499°  49'  =  -cot  49°  49       Reject  the  9 

etc.  etc.                           etc.  etc. 


16  FIELD  ASTRONOMY 

We  may  readily  formulate  a  corresponding  rule  for 
the  converse  process,  of  passing  from  the  function  to 
the  angle,  as  follows: 

(1)  When  the  arc  lies  in  an  odd  quadrant.     Look  out, 
in  the  first  quadrant,  the  angle  that  corresponds  to  the 
given  function  and  add  to  it  the  required  even  multiple 
of  90°,  i.e.,  o°  or  180°. 

(2)  When  the  arc  lies  in  an  even  quadrant.     Change 
the  name  of  the  function  (for  cos  read  sin,  for  tan  read 
cot,  etc.).     Look  out,  in  the  first  quadrant,  the  corre- 
sponding angle  and  add  to  it  the  required  odd  multiple 
of  90°,  i.e.,  90°  or  270°. 

See  the  following  examples,  in  which  we  represent 
the  required  angle  by  z  and  suppose  that  there  is  given 
the  numerical  value  of  its  tangent,  e.g.,  log  tan  2  =  9.654. 
The  process  of  looking  out  in  the  several  quadrants  the 
angle  corresponding  to  this  tangent  is  as  follows : 


Quadrant. 

Use. 

Angle. 

Add. 

Result. 

2d,  Even 

cot 

65°  45' 

90° 

155°  45' 

3d,  Odd 

tan 

24°   15' 

1  80° 

204°   15' 

4th,  Even 

cot 

65°  45'' 

270° 

335°  45' 

etc. 

etc. 

etc. 

etc. 

etc. 

The  degrees,  minutes,  and  seconds  of  the  required  angle 
should  be  obtained  from  the  table,  the  multiple  of  90° 
added  to  them,  and  the  final  result  written  down  in  its 
proper  place,  without  writing  the  intermediate  steps. 

7.  Determination  of  Angles.— In  this  connection  the 
student  will  do  well  to  examine  the  beginning  of  a  table 
of  logarithmic  trigonometric  functions  and  observe  how 
difficult  it  is  to  interpolate  accurately  the  value  of 
log  sin  z  or  log  tan  z,  corresponding  to  a  small  angle,  e.g., 


INTRODUCTORY.  17 

z  =  o°  33'  1 7".  The  difficulty  comes  from  the  rapid  varia- 
tion of  the  function,  large  and  changing  tabular  differ- 
ences. On  the  other  hand,  log  cos  z  changes  slowly  and 
may  be  readily  and  accurately  interpolated.  If  we  take 
the  converse  case  and  suppose  the  logarithmic  function 
to  be  given  and  the  corresponding  angle  required,  we 
shall  obtain  the  opposite  result.  The  angle  will  be  accu- 
rately determined  by  the  sine  or  tangent  and  very  poorly 
determined  by  the  cosine,  e.g.,  log  cos  o°  33'  17"  =9.99998 
and  every  angle  between  o°  29'  and  o°  36'  has  this 
cosine,  thus  leaving  a  possible  error  of  several  minutes 
in  the  value  of  the  angle  determined  from  this  function, 
while  the  log  sin,  if  correctly  given  to  five  decimal  places, 
will  determine  the  same  angle  within  a  small  fraction  of 
a  second. 

In  the  interest  of  precision  an  angle  should  always  be 
determined  from  a  function  that  changes  rapidly  (large 
tabular  differences),  while  a  quantity  that  is  to  be  found 
from  a  given  angle  is  best  determined  through  a  function 
that  changes  slowly.  In  the  example  of  §  2,  sin  a  might 
have  been  determined  through  sin  B  or  cos  B,  and  the 
former  was  used  for  this  purpose  because  it  varied  the 
more  slowly.  In  cases  of  this  kind,  and  they  are  very 
common,  use  the  function  that  stands  on  the  right-hand 
side  of  the  page,  in  the  tables,  and  subtract  it  from  the 
larger  of  the  two  numbers,  sin  a  sin  B  or  sin  a  cos  5,  and 
it  will  be  then  unnecessary  to  consider  whether  it  is  sine 
or  cosine  that  is  employed. 

The  angle  a,  in  this  example,  was  determined  through 
its  tangent  (log  tan  a  =  log  sin  a  — log  cos  a),  since  the 


18  FIELD  ASTRONOMY. 

tangent  always  varies  more  rapidly  than  either  sine  or 
cosine  and  should  generally  be  preferred  for  this  purpose. 
After  obtaining  a,  its  sine  and  cosine  were  looked  out 
from  the  tables  and  compared  with  the  numbers  obtained 
in  the  solution,  for  the  sake  of  the  "check"  thus  fur- 
nished upon  the  accuracy  of  the  numerical  work.  In 
subsequent  pages  other  checks  will  be  shown,  and  these 
should  be  applied  to  test  the  accuracy  of  numerical  work 
whenever  they  are  available.  The  mental  strain  accom- 
panying a  long  computation  is,  under  the  best  of  circum- 
stances, considerable,  and  a  check  properly  satisfied 
serves  to  relieve  this  tension  and  facilitate  the  subse- 
quent work. 

8.  Accuracy  of  Logarithmic  Computation. — The  exam- 
ple of  §  2  was  solved  with  logarithms  extending  only  to 
three-  places  of  decimals,  and  corresponding  to  this  use  of 
a  three-place  table  the  results  are  given  to  the  nearest 
tenth  of  a  degree.  If  it  were  required  to  obtain  results 
correct  to  the  nearest  minute  or  nearest  second,  a  greater 
number  of  decimals  must  be  employed  (four-,  five-,  or  six- 
place  tables).  The  labor  of  using  these  tables  increases 
very  rapidly  as  the  number  of  decimals  is  increased,  and 
a  compromise  is  always  to  be  made  between  extra  labor 
on  the  one  hand  and  limited  accuracy  on  the  other. 

As  the  choice  of  a  proper  number  of  decimal  places 
is  usually  an  embarrassing  one  for  the  beginner,  there  is 
given  below  for  his  guidance  a  formula  intended  to  repre- 
sent, at  least  approximately,  the  limit  of  error  to  be  ex- 
pected in  the  results  of  computation  on  account  of  the 
inherent  imperfections  of  logarithms  (neglected  deci- 


INTRODUCTORY.  19 

mals,  etc.V.  The  actual  error  may  fall  considerably 
short  of  this  limit  or  may  overstep  it  a  little.  It  is 
evident  that  the  limit  will  be  greater  for  a  long  compu- 
tation than  for  a  short  one,  and  if  we  measure  the  length 
of  a  computation  by  the  number,  n,  of  logarithms  that 
enter  into  it  and  represent  by  m  the  number  of  decimal 
places  to  which  these  logarithms  are  carried,  there  may 
be  derived  from  the  theory  of  probabilities  the  following 
expression,  in  minutes  of  arc,  for  the  limit  of  probable 
error: 

Limit  =  2800' .  \fn .  io~m. 

Applying  this  formula  to  the  example  of  §  2  we  may 
put  n  =  i6,  m  =  2ty  and  find  10'  as  the  limit  of  unavoid- 
able error;  corresponding  well  with  the  one-tenth  of  a 
degree  to  which  the  results  were  carried.  If  the  data 
were  given  to  the  nearest  minute  and  it  were  required 
to  preserve  this  degree  of  accuracy  in  the  results,  we 
should  write, 

_i'  =  28oo'  .  4.  io~w, 

and  solving,  find  w  =  4.o,  i.e.,  a  four-place  table  is  re- 
quired for  this  purpose. 

Let  the  student  verify  by  means  of  the  above  equa- 
tions the  following  precepts; 

To  obtain  Use 

Tenths  of  degrees  Three-place  tables. 

Minutes  Four-place  tables. 

Seconds  Five-  or  six-place  tables. 

Tenths  of  seconds  Seven-place  tables. 


30  FIELD  ASTRONOMY. 

If  the  results  are  to  be  expressed  in  linear  instead  of 
angular  measure,  the  limit  of  error  must  be  represented 
as  a  fractional  part  of  the  quantity,  x,  that  is  to  be 
determined,  and  corresponding  to  this  case  we  have 

Limit  =  o .  8  x .  \/n .  io~m. 

Corollary.  Do  not  attempt  to  obtain  from  a  table 
more  than  it  is  capable  of  furnishing;  e.g.,  do  not  inter- 
polate hundred ths  of  a  degree  in  the  example  of  §  2, 
and  in  connection  with  linear  quantities  do  not,  as  a  rule, 
interpolate  more  than  three  significant  figures  from  a 
three-place  table,  four  from  a  four-place  table,  etc. 

9.  Logarithmic  Tables. — There  exists  a  great  variety 
of  logarithmic  tables  of  different  degrees  of  accuracy, 
from  three  to  ten  places  of  decimals,  and  having  deter- 
mined the  number  of  decimal  places  required  in  a  given 
computation,  the  choice  among  the  corresponding  tables 
is  largely  a  matter  of  personal  taste.  The  beginner, 
however,  will  do  well  to  observe  the  following  rules  for 
distinguishing  good  tables  from  bad  ones: 

(A)  Wherever    the    tabular    differences    exceed    10, 
a   good   table   should   furnish   proportional   parts,    PP, 
in  the  margin  of  each  page,  so  that  the  logarithms  may 
be  interpolated  "in  the  head." 

(B)  The  tables  should  be  accompanied  by  tables  of 
addition  and  subtraction  logarithms.     For  an  explana- 
tion of  these,  their  purpose  and  use,  the  student  is  re- 
ferred to  the  tables  themselves,  but  we  note  here  that 
by  their  aid  the  example  of  §  2  might  have  been  much 


INTRODUCTORY.  21 

more  conveniently  solved,  as  is  illustrated  in  a  similar 
problem  in  §  15. 

The  most  generally  useful  tables  are  those  of  five 
decimal  places,  but  computers  find  it  to  their  advantage 
to  have  and  use  at  least  one  table  of  each  kind,  from  three 
to  six  or  seven  places.  In  the  examples  solved  in  the 
present  work  the  following  tables  have  been  used : 

Three-place,  Johnson.     New  York. 

Four-place,  Slichter.      New  York.     Gauss.    Berlin. 

Five-place,  Albrecht.    Berlin. 

Six-place,  Albrecht's   Bremiker.      Berlin. 

As  a  very  useful  supplement  to  the  logarithmic  tables 
a  slide-rule  and  the  extended  multiplication  tables  of 
Crelle  and  Zimmermann  are  highly  esteemed. 


CHAPTER   II. 

COORDINATES. 

10.  Fundamental  Concepts.  —  For  most  purposes  of 
practical  astronomy  the  stars  may  be  considered  as 
attached  to  the  sky,  i.e.,  to  the  blue  vault  of  the  heavens, 
which  is  technically  called  the  celestial  sphere,  and  is  re- 
garded as  of  indefinitely  great  radius  but  having  the 
earth  at  its  centre,  so  that  a  plane  passing  through  any 
terrestrial  point  intersects  this  sphere  in  a  great  circle, 
and  parallel  planes  passing  through  any  two  terrestrial 
points  intersect  the  sphere  in  the  same  great  circle. 

If  the  axis  about  which  the  earth  rotates  be  produced 
in  each  direction,  it  will  intersect  the  celestial  sphere  in 
two  points,  called  respectively  the  north  and  south  poles. 
If  a  plumb-line  be  suspended  at  any  place,  P,  on  the 
earth's  surface,  and  be  produced  in  both  directions,  it 
will  intersect  the  celestial  sphere,  above  and  below,  in 
the  zenith  and  nadir  of  the  place.  The  direction  thus 
determined  by  the  plumb-line  is  called  the  vertical  of  the 
place. 

The  figure  (shape)  of  the  earth  is  such  that  the  vertical 
of  any  place,  when  produced  downward,  intersects  the 
rotation  axis,  and  a  plane  may  therefore  be  passed  through 

22 


COORDINATES.  23 

this  axis  and  the  vertical.  This  plane,  by  its  intersection 
with  the  celestial  sphere,  produces  a  great  circle  which 
passes  through  the  poles,  the  zenith  and  nadir,  and  is 
called  the  meridian  of  the  place,  P. 

A  plane  passed  through  P  perpendicular  to  the  direc- 
tion of  the  vertical  produces  by  its  intersection  with  the 
sphere  the  horizon  of  P.  Any  plane  passing  through  the 
vertical  is  called  a  vertical  plane  and  produces  by  its  inter- 
section with  the  sphere,  a  vertical  circle.  That  vertical 
circle  whose  plane  is  perpendicular  to  the  meridian  is 
called  the  prime  vertical. 

With  exception  of  the  poles,  all  of  the  terms  above 
defined  depend  upon  the  direction  of  the  vertical,  and 
as  this  direction  varies  from  place  to  place  upon  the 
earth's  surface  each  such  place  has  its  own  meridian, 
horizon,  zenith,  etc.,  while  the  poles  of  the  celestial 
sphere  are  the  same  for  all  places. 

A  plane  passed  through  the  centre  of  the  earth  per- 
pendicular to  the  rotation  axis  produces  by  its  inter- 
section with  the  earth's  surface  the  terrestrial  equator, 
and  by  its  intersection  with  the  celestial  sphere  it  pro- 
duces the  celestial  equator. 

Owing  to  the  motion  of  the  earth  in  its  orbit  we  see 
anything  within  the  orbit  from  different  points  of  view 
at  different  seasons  of  the  year,  and  by  the  earth's  motion 
the  sun  is  thus  made  to  describe  an  apparent  path  among 
the  stars,  making  the  complete  circuit  of  the  sky  in  a 
year.  This  path  is  a  great  circle  intersecting  the  celestial 
equator  in  two  points  diametrically  opposite  to  each 
other,  and  that  one  of  these  points  through  which  the 


24  FIELD  ASTRONOMY. 

sun  passes  on  or  about  March  22  of  each  year,  is  called 
the  vernal  equinox. 

ii.  Systems  of  Coordinates.  —  Most  of  the  problems 
of  practical  astronomy  require  us  to  deal  with  the  appar- 
ent positions  and  motions  of  the  heavenly  bodies  as 
seen  projected  against  the  sky,  and  for  this  purpose 
there  are  employed  several  systems  of  coordinates  based 
upon  the  concepts  above  denned,  and  three  of  these 
systems  we  proceed  to  consider.  These  are  all  systems 
of  polar  coordinates  having  the  following  characteristics 
in  common: 

(1)  The  origin  of  each  system  is  at  the  centre  of  the 
celestial  sphere. 

(2)  Each  system  has  a  fundamental  plane  and  con- 
sists of  an  angle  measured   in  the  fundamental    plane; 
an  angle    measured    perpendicular  to  the  fundamental 
plane;  and  a  radius  vector.     The  first  of  these  angles 
is  frequently  called  the  horizontal  coordinate,  and  the 
second  the  vertical  coordinate,  of  the  system.     Latitudes 
and  longitudes  on  the  earth  furnish  such  a  system  of 
coordinates.     The    longitude    of    St.    Louis    (horizontal 
coordinate)  is  measured  by  an  angle  lying  in  the  plane 
of  the  equator,  which  is  the  fundamental  plane  of  this 
system.     The  latitude  of  St.  Louis  (vertical  coordinate) 
is  measured  by  an  angle  lying  in  a  plane  perpendicular 
to  the  equator,  and  the  radius  vector  of  St.  Louis  is  its 
distance  from  the  centre  of  the  earth,  which  latter  point 
is  taken  as  the  origin  of  coordinates. 

(3)  In    each    system    the    horizontal    coordinate    is 
measured  from  a  fixed  direction  in   the   fundamental 


COORDINATES.  25 

plane,  called  the  prime  radius,  through  360°.  The  ver- 
tical coordinate  is  measured  on  each  side  of  the  funda- 
mental plane  from  o°  to  90°. 

(4)  Those    vertical    coordinates   are   called   positive 
that  lie  upon  the   same  side  of  the  fundamental  plane 
with  the  zenith  of  an  observer  in  the  northern  hemi- 
sphere of  the  earth.     Those  that  lie  upon  the  opposite 
side  of  the  fundamental  plane  are  negative. 

(5)  It  is  frequently  convenient  to  measure  a  vertical 
coordinate  from  the  positive  half  of  a  line  perpendicular 
to  the  fundamental  plane  instead  of  from  the  funda- 
mental plane  itself,  (e.g.,  in  §  2  we  take  as  the  vertical 
coordinate  of  St.  Louis  its  distance  from  the  pole  instead 
of  from  the  equator) .     In  such  cases  this  coordinate  is 
always  positive  and  is  included  between  the  limits  o° 
and  1 80°.     If  h  represent  any  vertical  coordinate  meas- 
ured in  the  manner  first  described  and  z  be  the  corre- 
sponding coordinate  measured  in  the  second  way,  we 
shall  obviously  have  the  relation,  z  =  go°  —  h. 

The    several    systems    of    astronomical    coordinates 
differ  among  themselves  in  the  following  respects : 
(a)  Different  fundamental  planes  for  the  systems. 

(6)  Different  positions  of  the  prime  radii  in  the  fun- 
damental planes. 

(c)  Different  directions  in  which  the  horizontal  co- 
ordinates increase. 

The  data  which  completely  define  each  system  of 
coordinates  are  given  in  the  following'  table  together 
with  the  names  of  the  several  coordinates,  the  letters 
by  which  these  are  usually  represented,  and  the  point 


26  FIELD  ASTRONOMY. 

of  the  heavens,  called  the  pole  of  the  system,  toward 
which  the  positive  half  of  the  normal  to  the  fundamental 
plane  is  directed.  The  terms  east  and  west  are  used 
in  this  table  with  their  common  meaning,  to  indicate 
the  direction  toward  which  the  horizontal  coordinate 
increases.  The  letters  associated  with  the  several  co- 
ordinates are  conventional  symbols  that  should  be 
committed  to  memory. 

SYSTEMS  OF  COORDINATES. 


I. 

II. 

III. 

Horizon 

Equator 

Equator 

Prime  radius  points  toward  *  . 

Meridian 

Meridian 

Vernal  Equinox 

Horizontal    coordinates   increase 

West 

West 

East 

Zenith 

North  Pole 

North  Pole 

Name  of  horizontal  coordinate..  .  . 
Name  of  vertical  coordinate  .. 

Azimuth=.<4 
Altitude=A 

Hour  angle=/ 
Declination=S 

Right  Ascension  =  a 
Declination=S 

EXERCISES. — Let  the  student  define  in  his  own  language  the  several 
quantities  above  represented  by  the  letters  A,  t,  a,  h,  and  d. 

1 .  What  is  the  azimuth  of  the  north  pole  ? 

2.  What  do  the  hour  angle  and  altitude  of  the  zenith  respectively 
equal  ? 

3.  What  are  the  azimuths  of  the  prime  vertical? 

4.  What  are  the  declinations  of  the  points   in  which  the  horizon 
cuts  the  prime  vertical? 

5 .  Does  d  in  the  second  system  differ  in  any  way  from  d  in  the  third 
system  ? 

The  directions  of  the  prime  radius  as  above  denned 
for  systems  I  and  II,  are  ambiguous,  since  the  meridian 
cuts  the  fundamental  plane  of  each  of  those  systems 
in  two  points.  Either  of  these  points  may  be  used 
to  determine  the  direction  of  the  prime  radius,  but  in 
general  that  one  is  to  be  employed  which  lies  south  of 
the  zenith. 


COORDINATES.  27 

Let  the  student  show  the  relation  between  the  coordinates  furnished 
in  System  I  by  adopting  each  of  the  possible  positions  for  the  prime 
radius. 

12.  Uses  of  the  Three  Systems. — It  is  well  to  consider 
here,  very  briefly,  the  reasons  for  using  more  than  one 
system  of  coordinates,  and  the  relative  advantages  and 
disadvantages  of  these  systems. 

The  coordinates  of  System  I  are  well  adapted  to 
observation  with  portable  instruments,  e.g.,  an  engineer's 
transit,  since  the  horizon  is  more  easily  identified  with 
such  an  instrument  than  is  any  other  reference  plane, 
and  the  circles  of  the  instrument  may  be  made  to  read, 
directly,  altitudes  and  azimuths.  The  horizon  has  been 
defined  by  reference  to  the  direction  of  a  plumb-line, 
but  in  practice  a  spirit-level,  or  the  level  surface  of  a 
liquid  at  rest,  'are  more  frequently  used  to  determine 
its  position. 

System  I  possesses  the  disadvantage  that,  through 
the  earth's  rotation  about  its  axis,  both  the  altitude  and 
azimuth  of  a  star  are  constantly  changing  in  a  compli- 
cated manner,  and  in  this  respect  System  II  possesses 
a  marked  advantage.  Since  the  normal  to  its  funda- 
mental plane  coincides  with  the  earth's  axis,  rotation 
about  this  axis  has  no  effect  upon  the  vertical  coordi- 
nates, declinations,  which  remain  unchanged,  while  the 
horizontal  coordinates,  hour  angles,  increase  uniformly 
with  the  time,  15°  per  hour,  and  are  therefore  easily 
taken  into  account  and  measured  by  means  of  a  clock. 

Suppose  a  watch  to  have  its  dial  divided  into  twenty- four  hours, 
instead  of  the  customary  twelve.  If  this  watch  be  held  with  its  dial 
parallel  to  the  plane  of  the  equator,  the  hour  hand,  in  its  motion  around 


28  FIELD  ASTRONOMY. 

the  dial,  will  follow  and  keep  up  with  the  sun  as  it  moves  across  the 
sky.  If  the  watch  be  turned  in  its  own  plane  until  the  hour  hand 
points  toward  the  sun,  the  time  indicated  upon  the  dial  by  this  hand 
will  be  approximately  the  hour  angle  of  the  sun,  and  the  zero  of  the 
dial  will  point  toward  the  meridian,  i.e.;-?  south. 

Let  the  student  compare  the  ideal  case  above  considered  with  the 
following  rough  rule  sometimes  given  for  determining  the  direction 
of  the  meridian  by  means  of  a  watch  with  an  ordinary,  twelve-hour, 
dial:  Hold  the  watch  with  its  dial  as  nearly  parallel  to  the  plane  of  the 
equator  as  can  be  estimated.  (See  §  13  for  the  position  of  this  plane.) 
Revolve  the  watch  in  this  plane  until  the  hour  hand  points  toward 
the  sun,  and  the  south  half  of  the  meridian  will  then  cut  the  dial  mid- 
way between  the  hour  hand  and  the  figure  XII. 

A  further  advantage  is  gained  in  the  third  system 
of  coordinates,  since  here  the  prime  radius  shares  in 
the  apparent  rotation  of  the  celestial  sphere  about  the 
earth's  axis,  and  both  the  horizontal  and  vertical  coor- 
dinates are  therefore  unaffected  by  this  motion.  In- 
struments have  been  devised  for  the  measurement  of 
the  coordinates  in  each  of  these  systems,  but  we  shall 
be  mainly  concerned  with  those  that  relate  to  the  first 
system,  and  shall  consider  System  III  as  employed 
chiefly  to  furnish  a  set  of  coordinates  independent  of  the 
earth's  rotation  and  of  the  particular  place  upon  the 
earth  at  which  the  observer  chances  to  be.  These 
features  make  it  suited  to  furnish  a  permanent  record 
of  a  star's  position  in  the  sky,  and  it  is  so  used  in  the 
American  Ephemeris  (see  §  21)  and  other  nautical  alma- 
nacs, where  there  may  be  found,  tabulated,  the  right 
ascensions  and  declinations  of  the  sun,  moon,  planets, 
and  several  hundred  of  the  brighter  stars. 

13.  Relations  between  the  Systems  of  Coordinates. — 
A  problem  of  frequent  recurrence  is  the  transformation 
of  the  coordinates  of  a  star  from  one  system  to  another ; 


COORDINATES.  23 

indeed  most  of  the  problems  of  spherical  astronomy  are, 
analytically,  nothing  more  than  cases  of  such  trans- 
formation, and  as  an  introduction  to  these  problems 
we  shall  examine  the  relative  positions  of  the  funda- 
mental planes  and  prime  radii  of  the  several  systems. 

The  plane  of  the  equator  intersects  the  plane  of  the 
horizon  in  the  east  and  west  line,  and  the  angle  between 
the  two  planes  is  called  the  colatitude,  since  it  is  the 
complement  of  the  geographical  latitude  of  the  place 


FIG.  2. 

to  which  the  horizon  belongs.  The  latitude  is  commonly 
defined  as  the  angular  distance  of  any  place  from  the 
equator,  but  more  precisely  the  latitude  is  the  angle 
which  the  vertical  of  the  given  place  makes  with  the 
plane  of  the  equator.  From  Fig.  2,  which  represents 
a  meridian  section  of  the  earth  with  the  several  lines 


30  FIELD  ASTRONOMY. 

and  planes  passed  through  its  centre,  it  is  apparent 
that  the  latitude,  0,  equals  the  declination  of  the  zenith 
and  also  equals  the  altitude  of.  the  pole.  The  angular 
distance  of  the  zenith  from  the  pole  is  equal  to  the  co- 
latitude,  90°  —  0. 

The  second  and  third  systems  of  coordinates  have 
the  same  fundamental  plane,  and  their  relation  to  each 
other  is  therefore  determined  by  the  angle,  6,  between 
their  prime  radii.  Since  one  of  these  prime  radii  is 
directed  toward  a  fixed  point  of  the  heavens,  while  the 
other  lies  in  a  meridian  of  the  rotating  earth,  it  is  evident 
that  the  angle  6  is  continuously  and  uniformly  variable, 
at  the  rate  of  360°  in  twenty-four  hours.  Methods  of 
determining,  for  any  instant,  the  value  of  this  angle, 
which  is  called  the  sidereal  time,  will  be  given  hereafter. 
For  the  present  we  note  that  0  may  be  regarded  as  the 
horizontal  coordinate  of  the  vernal  equinox  in  the  second 
system,  or  as  the  horizontal  coordinate  of  the  meridian 
in  the  third  system,  and  correspondingly  we  may  define 
the  sidereal  time  as  either  the  hour  angle  of  the  vernal 
equinox  or  the  right  ascension  of  the  meridian. 

14.  Transformation  of  .Coordinates. — The  transforma- 
tion of  coordinates  from  the  first  to  the  second  system 
is  conveniently  made  by  means  of  the  ' '  astronomical 
triangle,"  i.e.,  the  spherical  triangle  formed  by  the 
zenith,  the  pole,  and  the  star  whose  coordinates  are 
to  be  transformed.  In  Fig.  3  this  triangle  is  marked 
by  the  letters  PZS,  P  indicating  the  position  of  the 
celestial  pole;  Z,  the  observer's  zenith;  and  5,  the 
apparent  place  of  the  star  as  seen  against  the  sky.  Imag- 


COORDINATES. 


31 


ine  the  triangle  projected  against  the  sky  and  the  three 
points  to  be  visible  in  their  true  positions. 

PZ  is  an  arc  of  a  great  circle  passing  through  the  pole 
and  zenith  and  must  therefore  be  a  part  of  the  observer's 
celestial  meridian,  and  in  Fig.  2  we  have  already  seen 
that  this  arc  of  the  meridian  is  equal  in  length  to  the 
complement  of  the  latitude,  90°— 0.  The  broken  line 


FIG.  3. — The  Astronomical  Triangle. 

HE'  in  the  figure,  is  an  arc  of  a  great  circle,  every  part 
of  which  is  90°  distant  from  Z.  But  the  great  circle 
90°  distant  from  the  zenith  is  the  horizon,  and  the  arc  >. 
HS  that  measures  the  distance  of  5  from'Ff?  must  be 
the  star's  altitude,  h,  and  the  side  SZ  of  the  astronom- 
ical triangle,  being  the  complement  of  this  arc,  is  equal 
to  go°  —  h.  In  like  manner,  EEf ',  drawn  90°  distant 
from  P,  is  an-  arc  of  the  celestial  equator;  the  a-c  SE, 
that  measures  the  distance  of  5  from  EE1 ',  is  the  star's 
declination,  d,  and  the  side  PS  of  the  triangle  equals 
go°-d. 


32  FIELD  ASTRONOMY. 

The  star's  hour  angle,  i.e.,  horizontal  coordinate 
lying  in  the  equator,  is  measured  by  the  arc  EE',  and 
this  arc,  by  a  theorem  of  spherical  geometry,  is  numer- 
ically equal  to  the  spherical  angle,  t,  included  between 
EP  and  E'P,  which  is  therefore  the  star's  hour  angle. 
In  like  manner  the  spherical  angle  SZE'  is  shown  to  be 
equal  to  the  star's  azimuth,  A,  and  the  angle  SZP  of 
the  astronomical  triangle  is  equal  to  180°  —  A.  The 
third  angle  of  the  triangle,  marked  q  in  the  figure,  is 
called  the  parallactic  angle. 

To  apply  to  the  astronomical  triangle  the  fundamental 
formulae  of  spherical  trigonometry  derived  in  §  i,  we 
replace  the  general  symbols  used  in  Equations  4  by 
the  particular  values  which  they  have  in  the  astronomical 
triangle,  as  follows  : 

a  =   o°-&  A=t 


and  introducing  these  values  into  Equations  4  we  obtain 
the  required  formulae  for  transforming  altitudes  and 
azimuths  into  declinations  and  hour  angles,  as  follows: 

cos  h  sin  A  =  +  cos  d  sin  t, 

cos  h  cos  A  =  —  cos  0  sin  d  +  sin  0  cos  d  cos  t,       (14) 
sin  h  =  +  sin  0  sin  d  +  cos  0  cos  d  cos  t. 

The  transformation  formulae  between  the  second  and 
third  systems  are  much  simpler.  In  Fig.  3  if  V  repre- 
sent the  position  of  the  vernal  equinox,  we  shall  have 
the  arc  VEf,  or  the  corresponding  spherical  angle  at  P, 
equal  to  the  sidereal  time,  6,  since  the  sidereal  time  is  the 


COORDINATES.  33 

hour  angle  of  the  vernal  equinox.  Similarly  the  arc 
VE  and  its  corresponding  angle  at  P  are  equal  to  the 
right  ascension  of  the  star,  and  from  the  figure  we  then 
obtain  the  required  relations, 

a  +  t  =  6,  d  =  d (i4a) 

The  transformation  between  the  first  and  third  systems 
is  best  made  through  the  second  system;  i.e.,  by  using 
both  groups  of  formulas  14  and  i^a. 

15.  Problem  in  Transformation  of  Coordinates.  —  At 
the  sidereal  time  i3h  22m  493.3  the  altitude  and  azimuth 
of  a  star  were  measured  at  a  place  in  latitude  43°  4'  36", 
as  follows:  h  =  6 1°  19'  36",  A  =253°  9'  42".  Required 
the  right  ascension  and  declination  of  the  star. 

The  required  transformation  formulae  may  be  obtained 
from  the  astronomical  triangle  in  the  same  manner  as 
Equations  14,  and  are: 

cos  d  sin  t  =  +  cos  h  sin  A , 
cos  d  cos  /  =  +  cos  h  cos  A  sin  0  +  sin  h  cos  0, 
sin  d  =  —  cos  h  cos  A  cos  0+ sin  h  sin  0 
a  =  d-t. 

Note  that  6,  which  occurs  only  in  the  last  equation,  is 
expressed  in  hours,  minutes,  and  seconds  of  time,  and 
that  it  is  customary  to  express  both  a  and  t  in  these  units, 
i5°  =  ih,  etc. 

A  convenient  form  for  the  numerical  operations 
involved  in  solving  these  equations  is  given  below,  and 
the  student  should  trace  it  through,  verifying  each  num- 
ber and  ascertaining  why  the  work  is  arranged  as  it  is. 


34 


FIELD  ASTRONOMY. 


Compare  and  contrast  this  solution  with  the  one  con- 
tained in  §  2.  The  difference  of  arrangement  is  largely 
due  to  the  introduction  here  of  addition  and  subtraction 
logarithms.  These  are  indicated  in  the  schedule  by  the 
words  Add,  Subtract,  and  it  is  to  be  especially  borne  in 
mind  that  the  addition  indicated  by  the  word  Add 
requires  a  subtraction  logarithm  when  one  of  the  given 
terms  is  itself  a  negative  .quantity,  etc.  The  schedule 
shows  the  algebraic  operation  required  by  the  formula, 
but  the  arithmetical  character  of  the  operation  is  altered 
by  the  presence  of  an  odd  number  of  negative  signs. 

SOLUTION. 


sin  A 

9  .98097?^ 

sin  h  cos  <j> 

9  .80676 

cos  h 

9.68108 

cos  h  cos  A  sin  <j> 

8.97740^ 

cos  A 

,9  .  46  1  gin 

Add 

Q-75974 

sin  <£ 

'9.83441 

cos  S  cos  t 

9-737J4 

sin  h 

9.94318 

(See  p.  6) 

9.88380 

cos  h  cos  A 

9.14299^ 

cos  d  sin  t 

9  .  66205?^ 

cos  <£ 

9.86358 

t 

3  1  9°  5  5'  44" 

cos  S 

9-85334 

sin  h  sin  (f> 

9-77759 

t  (time) 

2ihi9m428.  9 

cos  h  cos  A  cos  0 

9  .0065772- 

6 

I3h22m49«.3 

Subtract 

o  .06797 

a 

i6h  3m  6s.  4 

sin  8 

9-84556 

8 

44°  29'  12"  '• 

The  declination,  d,  is  obtained  both  from  sin  d  and 
cos  d,  and  the  agreement  of  the  two  values  is  a  '  check ' 
upon  the  accuracy  of  the  work.  The  *  indicates  that 
this  check  has  been  applied. 


CHAPTER  III. 

TIME. 

16.  In  astronomical  practice,  time  is  measured  by 
watches  and  clocks  that  differ  in  no  essential  respect 
from  those  in  common  use,  but  in  addition  to  the  com- 
mon system  of  time  reckoning,  astronomers  employ 
several  others,  of  which  we  shall  have  to  consider  the 
following : 

Sidereal  Time,  already  referred  to  in  §  13. 

True  Solar  Time,  which  is  frequently  called  Appar- 
ent Solar  Time. 

Mean  Solar  Time,  which  is  the  common  system  of 
e very-day  life. 

These  three  systems  possess  the  following  features 
in  common:  In  each  system  that  common  phrase  "the 
time  of  day"  means  the  hour  angle  of  a  particular  point 
in  the  heavens,  which  we  shall  call  the  zero  point  of  the 
system.  The  unit  of  time  is  called  a  day  and  is,  in  every 
case,  the  interval  between  consecutive  returns  of  the 
zero  point  to  a  given  meridian;  i.e.,  consecutive  transits 
of  a  given  meridian  past  the  same  zero  point.  This  unit 
is  subdivided  into  aliquot  parts  called  hours,  minutes, 
and  seconds.  Each  day  begins  at  the  instant  when  the 
zero  point  is  on  the  meridian;  i.e.,  on  the  upper  half  of 

35 


36  FIELD  ASTRONOMY. 

the  meridian  (noon)  in  astronomical  practice,  on  the 
lower  half  of  the  meridian  (midnight)  in  civil  affairs.  In 
astronomical  practice  the  hours  from  the  beginning  of 
the  day  are  reckoned  consecutively,  from  o  to  24;  in 
civil  practice  from  o  to  12,  and  then  repeated  to  12  again, 
with  the  distinguishing  symbols  A.M.  and  P.M.  In  con- 
sequence of  the  different  epochs  at  which  the  day  begins, 
the  astronomical  date  in  the  A.M.  hours  is  one  day  be- 
hind the  civil  date;  e.g., 

Civil  Time May  10,    5h  A.M.     equals 

Astronomical  Time  May    9,  iyh. 

In  the  P.M.  hours  the  dates  agree. 

Since  an  hour  angle  must  be  reckoned  from  a  deter- 
minate meridian,  this  meridian  must  be  specified  in  order 
to  make  ''the  time"  a  determinate  quantity,  and  this 
specification  of  the  meridian  should  be  included  in  the 
name  assigned  to  the  time;  e.g.,  Local  Time  denotes  the 
hour  angle  of  the  zero  point  reckoned  from  the  observer's 
own  (local)  meridian.  Greenwich  Time  is  the  hour  angle 
of  the  zero  point  reckoned  from  the  meridian  of  Green- 
wich. Standard  Time  is  the  hour  angle  of  the  zero  point 
reckoned  from  some  meridian  assumed  as  standard; 
e.g.,  in  the  United  States  and  Canada  the  meridians  75°, 
QO°,  105°,  and  120°  west  of  Greenwich  are  called  standard, 
and  Eastern,  Central,  Mountain,  and  Pacific  Standard 
Times  are  hour  angles  reckoned  from  these  meridians. 

A  like  practice  is  followed  in  the  use  of  the  term  Noon ; 
e.g.,  Washington  Noon  is  the  instant  at  which  the  zero 
point  is  in  the  act  of  crossing  the  meridian  of  Washington. 


TIME.  37 

17.  Longitude  and  Time.  —  We  have  introduced  above 
a  reference  to  the  time  at  different  meridians,  and  we  have 
now  to  note  that  since  ''the  time"  is  defined  as  an  hour 
angle,  it  is  evident  that  the  number  of  hours,  minutes, 
and  seconds  expressing  either  time  or  hour  angle  will 
depend  upon  the  meridian  from  which  the  latter  is  meas- 
ured. The  difference  between  the  hour  angles  reckoned 
from  two  different  meridians  will  equal  the  angle  between 
the  meridians,  i.e.,  their  difference  of  longitude,  so  that 
if  T'  and  T"  represent  the  times  of  any  event  (whether 
sidereal,  mean  solar,  or  true  solar  time)  referred  to  two 
different  meridians  whose  difference  of  longitude  is  A, 
we  shall  have  T1  —  T"  =  A.  It  is  customary  in  astronom- 
ical practice  to  express  differences  of  longitude  in  hours 
rather  than  in  degrees,  since  both  members  of  the  pre- 
ceding equation  should  be  given  in  terms  of  the  same 
units. 

By  transposition  of  one  term  in  the  preceding  equa- 
tion we  obtain 


(16) 


and  this  extremely  simple  equation  indicates  that  any 
given  time  referred  to  the  second  meridian  may  be  re- 
duced to  the  corresponding  time  of  the  first  meridian 
by  addition  of  the  difference  of  longitude,  where  this  dif- 
ference, A,  is  to  be  counted  a  positive  quantity  when  the 
second  meridian  is  west  of  the  first.  A  very  common 
blunder  is  to  omit  this  reduction  to  the  prime  meridian 
when  interpolating  from  the  almanac  (see  §  21).  Beware 
of  it,  and  note  that  the  hour  and  minute  for  which  a  quan- 


38  FIELD  ASTRONOMY. 

tity  is  required  to  be  interpolated  are  usually  given  in 
the  time  of  some  meridian  other  than  that  of  Greenwich 
or  Washington,  for  which  the  almanac  is  constructed, 
and  must  therefore  be  reduced  to  one  of  these  standard 
meridians,  by  addition  of  the  longitude,  before  they  can 
serve  as  the  argument  for  the  tabular  quantity  sought. 

1 8.  The  Three  Time  Systems. — The  several  time  sys- 
tems differ  one  from  another  chiefly  in  respect  of  their 
zero  points,  and  these  we  have  now  to  consider. 

Sidereal  Time. — As  already  indicated,  the  zero  point 
of  this  system  is  the  vernal  equinox,  and  since  this  is  a 
point  of  the  heavens  whose  position  with  respect  to  the 
fixed  stars  changes  very  slowly,  it  measures  well  their 
diurnal  motion.  In  colloquial  language,  ' '  the  stars  run 
on  sidereal  time, ' '  and  this  system  is  chiefly  used  in  con- 
nection with  their  apparent  diurnal  motion. 

Solar  Time. — As  their  names  indicate,  both  True 
Solar  Time  and  Mean  Solar  Time  have  zero  points  that 
depend  upon  the  sun,  and  before  drawing  any  distinction 
between  the  two  systems  we  recall  that,  owing  to  the 
earth's  annual  motion  in  its  orbit,  the  sun's  position 
among  the  stars  changes  from  day  to  day  (we  see  it  from 
different  standpoints).  While  this  change  in  the  sun's 
position  is  not  an  altogether  uniform  one  and  takes  place 
in  a  plane  inclined  to  that  of  the  earth's  rotation  (ecliptic, 
and  equator),  its  net  result  is  that  in  each  year  the  sun 
makes  one  entire  circuit  of  the  sky,  so  that  any  given 
meridian  of  the  earth,  in  the  course  of  a  year,  makes  one 
less  transit  over  the  sun  than  over  a  star,  or  over  the 
vernal  equinox.  The  number  of  solar  days  in  a  year  is 


TIME.  39 

therefore  one  less  than  the  number  of  sidereal  days;  e.g., 
for  the  epoch  1900,  (according  to  Harkness,) 

One  (tropical)  year  =  366. 242 197  sidereal  days 

=  365.242197  solar  days.    .      (17) 

It  appears  from  this  relation  that  a  sidereal  unit  of  time 
(day,  hour,  minute)  must  be  shorter  than  the  corre- 
sponding solar  unit,  a  relation  that  we  shall  have  to  con- 
sider hereafter. 

Apparent,  or  True,  Solar  Time. — This  system  has  for 
its  zero  point  the  centre  of  the  sun,  and  the  hour  angle 
of  the  sun's  centre  at  any  moment  is,  therefore,  the  true 
solar  time.  This  system  is  very  convenient  for  use  in 
connection  with  observations  of  the  sun,  but  owing  to 
the  irregularities  in  the  sun's  motion,  above  noted, 
apparent  solar  days,  hours,  etc.,  are  of  variable  length, 
a  day  in  December  being  nearly  a  minute  longer  than  one 
in  September.  When  time  is  to  be  kept  by  an  accurately 
constructed  clock  or  watch  such  irregularities  are  intoler- 
able, and  for  the  sake  of  clocks  and  watches  there  is 
employed  for  most  purposes  the  third  system,  viz., 

Mean  Solar  Time. — In  this  system  the  days  and  other 
units  are  of  uniform  length  and  equal,  respectively,  to  the 
mean  length  of  the  corresponding  units  of  apparent  solar 
time.  The  zero  point  of  the  system  is  an  imaginary 
body,  called  the  mean  sun,  that  is  supposed  to  move 
uniformly  along  the  equator,  keeping  as  nearly  in  the 
same  right  ascension  with  the  true  sun  as  is  consistent 
with  perfect  uniformity  of  motion.  The  mean  solar 
time  at  any  moment  is  the  hour  angle  of  the  mean  sun 


40  FIELD  ASTRONOMY. 

and,  numerically,  it  differs  from  the  corresponding  true 
solar  time  by  the  difference  between  the  hour  angles, 
or  right  ascensions,  of  the  true  and  mean  suns.  This 
difference  is  called  the  equation  of  time  (the  "sun  fast" 
and  ' '  sun  slow"  of  the  common  almanacs  and  calendars), 
and  its  value  for  each  day  of  the  year,  at  Greenwich  noon 
and  at  Washington  noon,  is  given  in  the  American 
Ephemeris  (see  §  21)  and  other  almanacs. 

To  change  local  solar  time  from  one  system  to  the  other 
we  have  therefore  to  interpolate  the  equation  of  time 
from  the  almanac,  with  the  argument  the  given  local 
time,  reduced  to  the  Greenwich  or  Washington  meridian 
by  addition  of  the  longitude,  and  apply  this  difference 
with  its  proper  sign  to  the  given  local  time.  For  exam- 
ple, let  it  be  required  to  find  for  the  meridian  of  Denver, 
and  for  the  date  May  10,  1905,  the  local  apparent  solar 
time  corresponding  to  the  mean  solar  time,  M,  given 
below.  The  course  of  the  computation  is  as  follows : 


< 


A,  Denver  west  of  Greenwich 6h  59™  47*. 6  ' 

M ,  Denver  Mean  Solar  Time 3      5     10  .5  * 

Greenwich  Mean  Solar  Time 10      4     58  .1  J  +  2 

Equation  of  Time +3    44  .2  3 

Denver  Apparent  Solar  Time 3      8    54  .7  *  +  3 

19.  Relation  of  Sidereal  to  Mean  Solar  Time.  —  Since 
the  sun  makes  the  complete  circuit  of  the  heavens  once 
in  each  year  it  must,  once  in  each  year,  have  the  same 
right  ascension,  and  therefore  the  same  hour  angle,  as 
the  vernal  equinox.  At  this  particular  moment,  which 
we  shall  represent  by  the  symbol  V,  sidereal  and  mean 
solar  time  will  agree,  but  at  any  other  moment  they  will 


TIME.  41 

differ,  since,  the  sidereal  units  of  time  being  shorter  than 
the  solar  ones,  sidereal  time  gains  continuously  and  uni- 
formly upon  mean  solar  time,  with  a  daily  rate  that  we 
may  represent  by  the  letter  a.  Let  D  represent  any 
given  moment  of  the  year,  and  let  6  and  M  be  the  corre- 
sponding sidereal  and  mean  solar  times;  we  shall  then 
have,  at  the  instant  D, 

6-M  =  a(D-V),  (18) 

where  the  interval  D  —  V  must  be  expressed  in  days, 
since  a  is  a  daily  gain.  This  daily  gain  is  clearly  equal 
to  the  difference  of  length  of  the  sidereal  and  solar  day, 
and  putting 

i  Sidereal  Day  =  i  Solar  Day  —  a", 
i  Solar  Day      =  i  Sidereal  Day  +  a'", 

where  a"  and  a'"  are  the  values  of  a  expressed  respectively 
in  solar  seconds  and  in  sidereal  seconds,  we  readily  find, 
from  Equation  17, 

a"  =  235s.9io,         a'"  =  2368.556.  (19) 

Where  only  a  rough  determination  of  the  difference 
between  sidereal  and  mean  solar  time  is  required  we  may 
assume  in  Equation  18, 

V  =  March  2  2. 6,  a  =  a'"=4m[i-TVL 
and  obtain, 

0=M  +  4m(£>- March  22.6)  [i-TV].  (20) 

This  formula  will  furnish  a  result  correct  within  one  or 
two  minutes. 


42  FIELD  ASTRONOMY. 

If  greater  precision  than  the  above  is  required  we 
must  use  more  accurate  values  of  V  and  a,  and  to  this 
end  there  is  given  in  the  American  Ephemeris,  for  Wash- 
ington (and  Greenwich)  Mean  Noon  of  every  day  of  the 
year,  the  value  of  the  term  a(D—  V),  which  is  there 
called  the  Sidereal  Time  of  Mean  Noon.  We  shall  repre- 
sent this  quantity  by  the  symbol  Q,  and  we  note  that  for 
any  other  time  than  noon,  or  any  other  meridian  than 
that  of  Washington,  the  difference  between  sidereal 
and  mean  solar  time,  6  and  M  respectively,  is  equal  to 
Q,  plus  the  gain  of  one  time  upon  the  other  in  the  interval 
from  Washington  mean  noon  to  the  given  6  or  M  ;  e.g.,  for 
a  place  whose  longitude  west  of  Washington  is  repre- 
sented by  A  we  have, 


In  this  equation,  for  any  given  place,  the  term  \a'"  is  a 
constant,  whose  value  may  be  determined  once  for  all 
and  written  in  the  margin  of  the  page  containing  the 
values  of  Q,  so  that  we  may  take  out  from  the  almanac 
for  any  given  day,  at  a  glance,  and  without  interpola- 
tion, instead  of  Q,  the  sum, 

"=QV  (21) 


where  Ql  is  for  the  local  meridian  what  Q  is  for  the  Wash- 
ington meridian.  We  shall  then  have  as  the  relation 
between  the  local  6  and  M, 


",  (22) 

* 

which  is  to  be  used  for  the  accurate  conversion  of  mean 
solar  into  sidereal  time. 


TIME.  43 

For  the  converse  process,  converting  sidereal  into 
mean  solar  time,  we  have  the  corresponding  relation 

where  the  last  term  is  the  equivalent  of  the  Ma'"  of  the 
preceding  equation,  but  is  expressed  in  sidereal  units. 
The  numerical  values  of  (0  —  Q^a"  and  Ma"'  are  most 
conveniently  to  be  obtained  from  Tables  II  and  III,  at 
the  end  of  the  almanac.  They  give  the  values  of  these 
terms  for  each  minute  and  second  of  the  twenty-four 
hours,  with  the  arguments  6  —  Q1  and  M,  expressed, 
respectively,  in  sidereal  and  mean  solar  time.  Take  Xa!" ', 
the  constant  correction  to  Q,  from  Table  III,  with  X  as 
the  argument. 

To  illustrate  the  actual  process  of  changing  time 
from  one  system  to  the  other  we  shall,  for  an  assumed 
date  given  below,  convert  the  Boston  mean  solar  time, 
9h  i9m  26S.67,  into  the  corresponding  sidereal  time,  and 
then  reconvert  this  sidereal  time  into  mean  solar  time. 
The  final  result  should,  of  course,  be  the  same  as  the 
initial  value  of  M.  The  difference  of  longitude  between 
Boston  and  Washington  is  assumed  to  be  X  =  —  oh  24™  is. 


Date 

1905,  Aug. 

4 

Q 

8  50  22.07 

Xa'" 

-3-95 

M 

9 

19 

26 

67 

6 

18 

ii 

16 

69 

J&n 

8 

5° 

18 

12 

8 

5° 

18 

12 

Ma'" 

31 

90 

( 

r1 

9 

20 

58 

57 

6 
(0) 

18 
18 

ii 
ii 

16 

69 

(6- 

M  a 

9 

—  i 

31 
26 

90 
67 

The  value  above  marked  (6),  is  that  found  from  the 
approximate  formula,  Equation  20,  and  is  given  for  com- 


44  FIELD  ASTRONOMY. 

parison  only.  It  forms  no  part  of  the  actual  conversion 
of  M  into  6. 

20.  Chronometer  Corrections.  —  As  already  indicated, 
in  actual  practice  the  measurement  of  time  is  made  by 
clocks  or  chronometers. 

A  chronometer  does  not  differ  essentially  from  an 
ordinary  watch,  and  like  the  latter  is  designed  to  show 
upon  its  face,  at  each  moment,  the  mean  solar  time  (or 
sidereal  time)  of  some  definite  meridian,  e.g.,  the  merid- 
ian 90°  west  of  Greenwich.  Since  the  time  indicated 
by  such  an  instrument  is  seldom  correct,  the  error  of  the 
timepiece  must  usually  be  taken  into  account,  and  in 
astronomical  practice  this  is  done  through  the  equation, 

0  =  T+JT,  or    M  =  T  +  AT,  (24) 

where  T  is  the  time  shown  by  the  chronometer  (or 
watch)  and  AT  is  the  correction  of  the  chronometer, 
i.e.,  the  quantity  which  must  be  added,  algebraically, 
to  the  watch  time  in  order  to  obtain  true  time  of  the 
given  meridian.  When  the  chronometer  is  too  slow  T 
is  less  than  the  true  time  at  any  moment,  and  AT  is  there- 
fore positive  in  this  case  and  negative  when  the  chro- 
nometer is  too  fast.  While  the  symbol  AT  always  repre- 
sents a  chronometer  correction  its  numerical  value  in  a 
given  case  depends  upon  the  particular  use  required, 
i.e.,  whether  the  chronometer  time  is  to  be  reduced  to 
sidereal,  or  solar,  local,  or  standard  time.  In  the  two 
Equations  24,  therefore,  AT  represents  quite  different 
quantities,  since  6  and  M  are  usually  different  one  from 
the  other,  and  in  every  case  a  special  memorandum 


TIME.  45 

must  be  made  showing  whether  the  given  AT  relates  to 
sidereal,  mean  solar,  or  apparent  solar  time. 

If  the  chronometer  gains  or  loses,  it  is  said  to  have  a 
rate  and  AT  will  then  change  from  day  to  day.  If  we 
assume  a  uniform  rate,  the  relation  between  T  and  0 
becomes 

e-T  +  JTt+p(T-TJ,  (25) 

where  the  subscript  0  denotes  the  particular  value  of  AT 
belonging  to  the  chronometer  time  TQ,  and  p  is  the  rate 
of  the  chronometer  per  day  or  per  hour,  positive  when 
the  chronometer  is  losing  time.  The  interval  T—T0 
must  be  expressed  in  the  same  unit  as  that  for  which  p 
is  given,  hours  for  an  hourly  rate,  etc.  A  similar  equa- 
tion represents  the  relation  between  T  and  M,  but  p  and 
AT  will  be  numerically  different  from  the  values  required 
in  Equation  25. 

A  sidereal  chronometer,  i.e.,  one  intended  to  keep 
sidereal  time,  differs  from  a  mean  solar  chronometer 
only  in  the  more  rapid  motion  of  its  mechanism,  and  is 
in  fact  an  ordinary  timepiece  for  which  p=—  3m  56S.5 
per  day.  Similarly  a  watch  may  be  regarded  as  a 
sidereal  timepiece  for  which  p=  +  ios  per  hour.  A 
sidereal  chronometer  is  most  convenient  for  use  in  obser- 
vations of  stars,  since  their  diurnal  motion  in  hour  angle 
is  proportional  to  the  lapse  of  sidereal  time,  but  these 
observations  may  perfectly  well  be  made  with  a  watch 
or  other  mean  solar  timepiece,  provided  this  is  treated 
as  a  sidereal  chronometer  with  a  large  rate;  e.g.,  if  p 
denote  the  hourly  rate  of  the  watch  relative  to  mean  solar 


46  FIELD  ASTRONOMY. 

time,  its  hourly  rate  upon  sidereal  time  will  be  p'  =p+  io8. 
Use  Equation  25  in  connection  with  this  value  of  pr 
to  determine  from  minute  to  minute  the  varying  value 
of  J7. 

21.  The  Almanac.  —  The  American  Ephemeris  and 
Nautical  Almanac  is  an  annual  volume  issued  by  the 
U.  S.  Navy  Department  for  the  use  of  navigators,  astron- 
omers, and  others  concerned  with  astronomical  data. 
These  data  are  for  the  most  part  quantities  that  vary 
from  day  to  day  and  whose  numerical  values  are  given 
at  convenient  intervals  of  Greenwich  or  Washington 
solar  time,  e.g.,  the  E  and  Q  of  the  preceding  sections, 
and  the  right  ascensions  and  declinations  of  the  sun, 
moon,  planets,  and  principal  fixed  stars.  The  varia- 
tions of  these  quantities  are  due  to  many  causes,  orbital 
motion,  precession,  nutation,  aberration,  etc.,  that,  in 
general,  lie  beyond  the  scope  of  the  present  work,  but 
we  shall  have  frequent  occasion,  as  in  §§  18  and  19,  to 
take  from  the  almanac  numerical  values  of  the  quanti- 
ties above  indicated,  and  these  values  are  to  be  inter- 
polated for  some  particular  instant  of  time,  usually  that 
of  an  observation  in  connection  with  which  they  are 
required,  as  logarithms  are  interpolated  to  correspond 
to  some  particular  value  of  the  argument  of  the  table. 
Since  quantities  are  tabulated  in  the  almanac  for  selected 
instants  of  Greenwich  or  Washington  time,  the  time  used 
as  the  argument  for  their  interpolation  must  be  referred 
to  one  of  these  meridians  (see  §  17). 

For  a  detailed  account  of  the  way  in  which  the  almanac 
is  to  be  used,  consult  the  explanations  given  at  the  end  of 


TIME.  47 

each  volume,  under  the  title,  Use  of  the  Tables.  In 
addition  to  those  explanations  it  should  be  noted  that 
under  the  heading  Fixed  Stars,  pages  304-399,  there 
are  given  three  separate  tables,  from  the  last  of  which, 
bearing  the  subtitle  Apparent  Places  for  the  Upper 
Transit  at  Washington,  accurate  coordinates  of  most  of 
the  stars  may  be  obtained  for  use  in  the  reduction  of 
observations.  For  the  remaining  stars,  five  in  number 
and  all  very  near  the  celestial  pole,  special  provision  of 
this  kind  is  made  in  the  second  table,  which  bears  the 
subtitle  Circumpolar  Stars.  Look  here  for  the  coor- 
dinates of  Polaris.  The  first  table,  under  the  subtitle 
Mean  Places,  etc.,  gives  in  very  compact  form,  for  all 
stars  contained  in  the  other  two  tables,  right  ascensions 
and  declinations,  together  with  their  Annual  Variations, 
that  may  be  consulted  with  advantage  when  only  approx- 
imate values  of  these  quantities  are  required,  e.g.,  in  the 
preliminary  selection  of  stars  suitable  to  be  observed. 

In  this  connection  the  second  column  of  the  table 
of  Mean  Places,  entitled  Magnitude,  deserves  especial 
notice,  since  it  furnishes  an  index  to  the  brightness  of 
the  stars,  which  is  an  important  element  in  deciding 
upon  their  availability  for  a  given  instrument.  The 
brightness  of  each  star  is  represented  by  a  number 
adapted,  upon  an  arbitrary  scale,  to  that  brightness, 
so  that  a  very  bright  star  is  represented  by  the  number 
o,  one  at  the  limit  of  naked  eye  visibility  by  6, 
and  intermediate  degrees  of  brightness  are  represented 
by  the  intermediate  numbers,  carried  to  tenths  of  a  mag- 
nitude. Polaris  is  of  the  magnitude  2.2,  and  is  a  con- 


48  FIELD  ASTRONOMY. 

spicuous  object  in  even  a  very  small  telescope,  provided 
the  telescope  is  properly  focussed.  In  the  telescope 
of  an  engineer's  transit,  stars  of  ythe  magnitude  4.0  or 
even  5.0  may  be  readily  observed,  while  with  a  sextant, 
under  ordinary  conditions,  the  third  magnitude  may 
be  taken  as  the  limit  of  availability. 


CHAPTER  IV. 
CORRECTIONS  TO   OBSERVED   COORDINATES. 

IT  has  already  been  pointed  out  that  the  problems 
of  practical  astronomy  are  in  great  part  cases  of  the 
transformation  of  coordinates  between  systems  having 
a  common  origin  but  different  axes,  and  it  should  be 
noted  that  the  observed  data  for  these  transformations 
frequently  require  some  correction  before  they  can  be 
introduced  into  the  equations  furnished  by  the  astro- 
nomical triangle.  Aside  from  errors  arising  from  de- 
fective adjustment  or-  other  purely  instrumental  causes, 
the  observed  coordinates  of  a  celestial  body  may  require 
any  or  all  of  the  following  corrections. 

22.  Dip  of  the  Horizon.  —  This  correction  is  required 
when  an  altitude  is  to  be  derived  from  a  measurement 
of  the  angle  of  elevation  of  a  body  above  the  sea  horizon. 
Owing  to  the  spherical  shape  of  the  earth  the  visible 
sea  horizon  always  lies  below  the  plane  of  the  observer's 
true  horizon,  and  the  amount  of  this  depression  might 
easily  be  determined  from  the  geometrical  conditions 
involved,  were  it  not  that  the  rays  of  light  coming  to 
the  observer  from  near  the  horizon  are  bent  by  the  at- 
mosphere (refraction),  in  a  manner  that  does  not  admit 

49 


QorTHE<r"\ 
'N/VERS/TY  ) 

s 


50  FIELD  ASTRONOMY. 

i 

of  accurate  estimation  in  any  given  case,  although  its 
average  amount  is  fairly  well  known.  We  therefore 
abstain  from  any  formal  investigation  of  this  correc- 
tion, and  expressing  by  e,  in  feet,  the  observer's  elevation 
above  the  water,  we  adopt  as  a  sufficient  approximation 
to  the  observed  amount  of  the  depression,  either  of  the 
following  formulae, 

£>'  =  V*-TW*,       £>"  =  [i.7738]V*.  (26) 

The  values  of  D  given  by  these  equations  are  expressed 
in  minutes  and  seconds,  respectively,  but  owing  to  varia- 
tions in  the  amount  of  the  refraction  the  numerical 
values  furnished  in  a  given  case  may  be  in  error  by 
several  per  cent.  As  a  correction  D  must  always  be 
so  applied  as  to  diminish  the  observed  elevation  above 
the  horizon. 

Note  that  if  the  depression  of  the  visible  horizon 
be  measured  with  a  theodolite  or  other  suitable  instru- 
ment, Equation  26  will  furnish  an  approximate  value 
of  the  elevation  of  the  instrument  above  the  water. 

23.  Refraction.  —  In  general  the  apparent  direction 
of  a  star  is  not  its  true  direction  from  the  observer,  since 
the  light  by  which  he  sees  it  has  been  bent  from  its 
original  course  in  passing  through  the  earth's  atmos- 
phere. The  resulting  displacement  of  the  star  from  its 
true  position  is  called  refraction,  and,  like  the  similar 
effect  noted  in  the  previous  section,  its  analytical  treat- 
ment presents  mathematical  and  physical  problems 
whose  solution  must  be  sought  in  more  advanced  works 
than  the  present.  Some  of  the  results  of  that  solution 


CORRECTIONS   TO  OBSERVED  COORDINATES.          51 

which  we  shall  have  occasion  to  use  hereafter  are  as 
i'ollows:  Save  at  very  low  altitudes,  less  than  10°,  the 
refraction  does  not  sensibly  change  the  azimuth  of  a 
star,  but  its  whole  effect  is  to  increase  the  altitude,  so 
that  every  star  appears  nearer  to  the  zenith  than  it  would 
appear  if  there  were  no  refraction.  The  amount  of  this 
displacement  depends  chiefly  upon  the  star's  distance 
from  the  zenith,  but  is  also  dependent  in  some  measure 
upon  the  temperature  of  the  air  and  its  barometric 
pressure. 

If  we  represent  by  z'  the  star's  apparent  zenith  dis- 
tance, as  affected  by  refraction,  by  z  the  corresponding 
true  zenith  distance,  by  t  the  temperature,  in  degrees 
Fahr.,  of  the  air  surrounding  the  observer,  and  by  B 
the  barometric  pressure  in  inches,  the  amount  of  the 
refraction,  in  seconds  of  arc,  will  be  furnished  by  the 
following  two  equations,  which  for  all  altitudes  greater 
than  15°  faithfully  reproduce  the  refractions  of  the 
Pulkowa  Refraction  Tables,  and  furnish  values  that  may 
be  relied  upon  to  within  a  small  fraction  of  a  second 
of  arc. 


log  F  =  [4.079  -io](353°  +  0  tan2  sf, 

tans'  (27) 


For  most  purposes  of  field  astronomy  the  first  of  these 
equations  may  be  suppressed  and  the  divisor,  F,  be  put 
equal  to  unity;  the  resulting  error  in  the  computed 
refraction  will  rarely  be  greater  than  i". 

The  readings  of  a  mercurial  barometer,  B'  ,  do  not 


52  FIELD  ASTRONOMY. 

furnish  immediately  the  barometric  pressure,  B,  but 
require  a  "reduction  to  the  freezing-point,"  i.e.,  a  cor- 
rection to  reduce  the  reading  to  what  it  would  be  if  the 
mercury  were  at  the  normal  temperature  assumed  in  the 
theory  of  the  barometer.  This  reduction  may  be  ob- 
tained with  sufficient  accuracy  from  the  equation 


IO  OOO 


where  T  is  the  temperature  of  the  mercury,  in  degrees 
Fahr.,  and  the  barometer  reading  and  its  resulting  cor- 
rection are  expressed  in  inches. 

24.  Semi-diameter.  —  Observations  of  the  sun  or  other 
body  presenting  a  sensible  disk  are  usually  made  by 
pointing  the  instrument  at  the  edgo  of  the  body,  techni- 
cally called  the  limb,  and  the  resulting  altitude  or  azimuth 
is  that  of  the  limb  observed,  while  the  data  furnished 
by  the  almanac  relate  to  the  centre  of  the  body.  The 
semi-diameters  of  the  sun,  moon,  and  planets,  i.e.,  the 
angles  subtended  at  the  earth  by  their  respective  radii, 
are  given  in  the  almanac  at  convenient  intervals  of  time, 
and  the  interpolated  values  of  these  quantities  may  be 
used  to  pass  from  the  observed  coordinates  of  the  limb 
to  those  of  the  centre  of  the  body,  e.g.,  the  sun.  In 
the  case  of  the  altitude  or  zenith  distance  we  have  the 
very  simple  relation 


(29) 

where  S  denotes  the  semi-diameter  and  h"  and  hf  are,  re- 
spectively, the  observed  and  the  corrected  altitude.    The 


CORRECTIONS  TO  OBSERVED  COORDINATES. 


53 


sign  of  5  depends  upon  whether  the  lower  or  the  upper 
limb  was  observed. 

In  the  case  of  an  azimuth  the  relation  is  more 
complicated.  From  the  right-angled  spherical  triangle 
formed  by  the  zenith,  the  sun's  centre  and  that  point  of 
the  limb  at  which  the  latter  is  tangent  to  a  vertical  circle 
(see  Fig.  4)  we  obtain, 


sin  z  sin  (A'  —  A")  =  sin  5, 


(3°) 


which  determines  the  correction,  Af  —  A" ,  for  difference 
of  azimuth  between  centre  and  limb.     Since  5  does  not 
much  exceed    15',  we  may  in  most 
cases  assume  the  arcs  to  be  propor- 
tional to  their  sines  and  simplify  this 
rigorous  equation  to  the  form, 

A' =  A" ±5  sec/*, 

in  which  the  positive  sign  is  to  be 
used  for  the  following  and  the  nega- 
tive for  the  preceding  limb. 

25.  Parallax  —  In  the  reduction 
of  astronomical  observations  it  is 
usually  necessary  to  combine  the 
observed  coordinates,  azimuth,  alti- 
tude, etc.,  with  data  obtained  from 
the  almanac,  e.g.,  the  right  ascension  and  decima- 
tion of  the  body  observed.  But  the  origin  to  which 
these  latter  coordinates  are  referred  is  the  centre  of  the 
earth,  while  the  origin  for  the  observed  coordinates  is 


FIG.  4.— Semi-diameter. 


54 


FIELD  ASTRONOMY. 


at  the  eye  of  the  observer,  and  before  combining  these 
heterogeneous  data  we  must  reduce  them  to  a  common 
origin,  for  which  we  select  that  used  in  the  almanac. 

In  Fig.  5  let  C  represent  tjie  centre  of  the  earth,  0 
the  observer's  position,  and  P  the  observed  body,  at  the 
respective  distances  p  and  r  from  C.  Neglecting  the 


> 


FIG.  5.— Parallax. 

earth's  compression,  i.e.,  its  slight  deviation  from  a  truly 
spherical  form,  the  line  OC  is  the  observer's  vertical 
and,  therefore,  OPC  is  a  vertical  plane  and  marks  out 
upon  the  celestial  sphere  a  vertical  circle,  against  which 
the  body  P  will  appear  projected  whether  seen  from  0 
or  C.  Its  azimuth  will,  therefore,  be  the  same  for  the 
two  origins  and  requires  no  reduction  to  the  centre  of  the 
earth. 

The  altitude,  however,  does  require  such  a  reduction, 
and  to  determine  its  amount  we  let  OH  in  Fig.  5  repre- 
sent the  plane  of  the  observer's  horizon  and  obtain  as 


CORRECTIONS  TO  OBSERVED  COORDINATES.    55 

the  observed  altitude  of  P  the  angle  there  marked  hr . 
As  seen  from  the  centre  of  the  earth  the  altitude  of  P 
will  be  measured  by  the  angle  PIH,  marked  h  in  the 
figure,  and  from  principles  of  elementary  geometry  we 
have 

k-h'+tOPC. 

This  last  angle  is  called  the  parallax  in  altitude,  and  rep- 
resenting it  by  P  we  find  from  the  triangle  OPC 

p  sin  (90°  +  h')  =r  sin  P. 

Since  r  is  always  much  greater  than  p,  P  must  be  a  small 
angle  and,  applying  the  principles  of  §  4,  we  may  write, 
in  place  of  the  preceding  equation, 

P=h-h' =  20626$ -?-cosh',  (32) 

T 

which  is  the  required  correction  to  reduce  an  observed 
altitude  to  the  corresponding  coordinate  referred  to  an 
origin  at  the  centre  of  the  earth. 

For  the  fixed  stars  this  correction  is  absolutely  insen- 
sible, less  than  o".oooi,  on  account  of  their  great  distance 
from  the  earth.  For  the  sun  and  planets  it  amounts  to  a 
few  seconds  of  arc,  and  in  its  computation  the  value  of  the 

coefficient,  206265  — »  should  be  taken  from  the  almanac, 

where  it  is  given  for  each  of  these  bodies  and  is  called 
their  horizontal  parallax,  since  it  is  the  amount  of  the 
parallax  in  altitude  when  the  body  is  in  the  horizon, 
&'=o°.  For  the  sun  it  is  usually  sufficient  to  assume 


56  FIELD  ASTRONOMY. 

8".  8  as  a  constant  value  for  its  horizontal  parallax. 
The  moon's  parallax  is  much  greater,  about  i°,  and  the 
simple  analysis  given. above  neglects  some  factors  that 
are  of  sensible  magnitude  in  this  case,  although  for  ordi- 
nary purposes  they  may  be  ignored  in  connection  with 
every  other  celestial  body. 

Since  the  effect  of  parallax  is  to  make  the  body  appear 
farther  from  the  zenith  than  it  really  is,  the  corrections 
for  parallax  and  refraction  will  always  have  opposite 
signs. 

26.  Example. — In  the  application  of  the  several  cor- 
rections required  by  an  observed  altitude  they  should 
be  applied  in  the  order  in  which  they  have  been  treated 
above,  and  each  successive  partially  corrected  altitude 
should  be  used  as  the  value  of  h  required  in  computing 
the  next  correction.  As  an  example  of  such  corrections 
we  take  the  following  observed  angle  between  the  sun's 
upper  limb  and  a  water  horizon  as  seen  by  an  observer 
at  an  elevation  of  63  feet  above  the  water.  The  data 


REDUCTION    OF    AN    OBSERVED    ALTITUDE. 


Temp,  t 

44°  Fahr. 

Const. 

2.9922 

Barometer 

29.26  inches 

B 

1-4657 

Red.  to  freezing 

—  0.04 

colog  (456+0 

7.3010 

B 

20  .  22 

colog  F 

9.9983 

Observed  Angle 

27°  43'  20" 

tan  z' 

0.2818 

log  Refraction 

2.0390 

Dip.  Const. 

1.7738 

Refraction 

-i'   49"  -4 

\/e 

0.8996 

h" 

27°  33'  39"-2 

Dip 

47i".4=7'5i".4 

Semi-diam: 

—  16       i  .7 

k' 

27    35'  28".6 

h"' 

27  i7    37  -5 

Const. 

4.079 

cos  W" 

9-949 

353+' 

2-599 

8.8 

0.944 

tan2  sf 

0.564 

Parallax 

+  7".8 

log  (log  F) 

7.242 

True/t 

27°i7'45"-3 

CORRECTIONS   TO  OBSERVED  COORDINATES.          57 

furnished  directly  by  the  observation  are  placed  at  the 
top  of  the  first  column.  The  tan  zf  used  above  is  of 
course  equal  to  cot  hf  and  was  taken  from  the  logarith- 
mic tables  as  a  cotangent. 

In  accordance  with  general  custom  the  symbol  log 
is  printed  in  the  above  schedule  only  when  necessary  to 
avoid  misunderstanding,  as  at  the  bottom  of  the  first 
column.  Usually  the  figures  themselves  indicate  whether 
they  are  logarithms  or  natural  numbers;  e.g.,  the  several 
numbers  marked  Const,  are  clearly  the  logarithms  of 
constant  coefficients.  For  similar  reasons  of  convenience 
the  —  10  that  strictly  should  be  placed  after  a  logarithm 
whose  characteristic  has  been  increased  by  10  is  usually 
left  to  the  imagination. 

27.  Diurnal  Aberration.  —  There  is  a  very  small  cor- 
rection to  observed  data,  arising  from  the  fact  that  the 
observer  himself  is  not  at  rest  relative  to  the  stars,  but 
is  always  in  rapid  motion  toward  the  east  point  of  his 
horizon,  carried  along  by  the  earth  in  its  diurnal  rotation. 
This  correction  is  so  small  that  it  may  usually  be  omitted 
and  we  therefore  abstain  from  an  analytical  investigation 
of  its  effect,  such  as  may  be  found  in  the  larger  treatises 
upon  spherical  astronomy,  and  note  as  a  result  of  that 
investigation  that  all  stars  when  near  the  meridian  are 
displaced  toward  the  east  point  of  the  horizon  through 
an  angular  distance  equal  to  0^.32  cos  0,  where  0  de- 
notes the  observer's  latitude.  As  a  result  of  this  dis- 
placement each  star  comes  a  little  later  to  the  meridian 
than  it  otherwise  would  come  and  since  the  rate  of  motion 
of  a  star  when  measured  in  arc  of  a  great  circle  is  proper- 


58  FIELD  ASTRONOMY. 

tional  to  the  cosine  of  its  declination,  the  amount  of  this 
retardation,  expressed  in  time,  is  os.02i  cos  0  sec  d. 
See  the  theory  of  the  transit  instrument  for  an  example 
of  the  application  of  this  correction,  and  see  also  the  de- 
termination of  precise  azimuths  for  another  case  in  which 
diurnal  aberration  is  to  be  taken  into  account. 


CHAPTER  V. 


ROUGH    DETERMINATIONS    OF  TIME,  LATITUDE,  AND 

AZIMUTH. 


28.  General  Considerations.  —  For  the  purposes  of 
field  astronomy,  which  are  the  only  ones  contemplated 
in  the  present  work,  the  most  important  astronomical 
problems  relate  to  the  determination  of  time,  latitude, 
and  azimuth. 

A  time  determination  implies  the  making  and  reduc- 
ing of  astronomical  observations  which  suffice  to  furnish 
the  correction,  AT,  of  a  chronometer  or  other  timepiece, 
and  for  this  purpose  we  obtain  from  §§15  and  20  the 

relations 

a  +  t  =  6  =  T  +  JT,  (33) 

where  a  and  t  represent  the  right  ascension  and  hour  angle 
of  any  star  at  the  chronometer  time  T.  The  student 
should  particularly  note  that  the  chronometer  is  not 
supposed  to  be  correctly  set  ;  T  is  the  time  shown  by  the 
chronometer  regardless  of  whether  that  time  be  right  or 
wrong,  since  the  AT  fully  compensates  for  any  error  of 
this  kind.  In  the  case  of  the  sun  we  have,  from  the  rela- 
tion between  mean  and  apparent  solar  time, 

(34) 


where  E  denotes  the  equation  of  time  at  the  instant  T. 

59 


60  FIELD  ASTRONOMY. 

Since  a  and  E  may  be  obtained  from  the  almanac,  any 
observation  which  determines  the  hour  angle  of  a  celestial 
body  at  the  observed  time  T  w.ill  suffice  to  determine 
JT,  and  such  an  observation  when  properly  reduced 
constitutes  a  time  determination. 

An  azimuth  determination  may  be  required  either 
for  fixing  the  true  azimuth  of  the  line  joining  two  terres- 
trial points,  or  for  determining  the  relation  of  a  particular 
instrument  to  the  meridian;  e.g.,  to  determine  the  read- 
ing, K,  to  which  the  azimuth  circle  of  a  theodolite  must 
be  set,  in  order  that  the  line  of  sight  shall  point  due  south. 
A  theodolite  is  said  to  be  oriented  when  its  verniers  have 
been  set  to  read  the  true  azimuth  of  the  object  toward 
which  the  line  of  sight  is  directed,  i.e.,  when  K  =  o. 

By  a  latitude  determination  we  mean  any  set  of  ob- 
servations from  which  a  knowledge  of  the  observer's 
latitude  may  be  obtained. 

For  each  of  these  determinations,  time,  azimuth, 
latitude,  many  methods  have  been  devised  and  these 
differ  greatly  among  themselves  with  respect  to  the 
instrumental  equipment  and  expenditure  of  time  and 
labor  which  they  require,  and  with  respect  to  the  corre- 
sponding degree  of  accuracy  furnished  in  their  results. 
In  any  given  case  a  choice  must  be  made  among  these 
methods  with  reference  to  the  required  precision  of  the 
results  and  also  with  reference  to  convenience  and 
economy  in  obtaining  it.  To  facilitate  this  choice  the 
methods  to  be  presented  in  the  following  pages  are 
classified  as: 

(A)  Rough  Determinations;  in  which  there  may  be 


ROUGH  DETERMINATIONS  OF  TIME,  ETC.  61 

permitted  in  the  final  result  an  error  amounting  to  two 
minutes  of  arc  or  one  tenth  of  a  minute  of  time. 

(B)  Approximate  Determinations;  in  which  the  final 
errors  ought  not  to  exceed  15"  and  is  respectively. 

(C)  Accurate  Determinations;  in  which  the  required 
precision  is  limited  only  by  the  capacity  of  the  instru- 
ment and  of  the  observer.   In  the  case  of  a  sextant  this 
limit  may  be  placed  at  2"  or  3",  and  for  a  good  engineer's 
transit  at  i".     We  proceed  first  to  consider  that  class 
of   observations  whose   advantage  consists  in  economy 
of  time  and  labor,  viz.,  rough  determinations. 

29.  Latitude.  —  A  determination  of  any  one  of  the 
quantities  time,  latitude,  or  azimuth  is  greatly  facilitated 
by  a  knowledge  of  one  or  both  of  the  others,  and  if  all 
three  are  unknown,  the  simplest  mode  of  procedure  is 
to  observe  the  Pole  Star  as  set  forth  in  §  32.  But  this 
commonly  requires  observations  by  night,  which  may 
be  inconvenient,  and  by  day  the  sun  is  the  object  most 
readily  available. 

From  the  astronomical  triangle,  or  from  Equations  14, 
it  is  apparent  that  when  the  sun  is  on  the  meridian, 
i.e.,  when  t  =  o,  its  altitude  is  a  maximum,  and  if  this 
maximum  altitude  be  measured  with  a  sextant  or  theod- 
olite it  will  furnish  a  latitude  determination  through  the 
equation 

h,  (35) 


which  may  be  obtained  -by  inspection  from  Fig.  2,  or 
analytically  from  the  last  of  Equations  14.  With  the 
instrument  employed,  follow  the  sun's  motion  in  altitude 


62  FIELD  ASTRONOMY. 

until  it  begins  to  diminish,  and  take  the  greatest  reading 
ootained  as  corresponding  to  the  maximum  altitude. 

This  reading,  or  the  altitude,  h',  derived  from  it,  will 
require  correction  for  instrumental  errors,  semi-diameter, 
etc.,  as  shown  in  Chapter  IV,  but  the  application  of  these 
corrections  may  be  abbreviated  by  interpolating,  in 
minutes  of  arc,  the  combined  correction  for  refraction  and 
parallax  from  the  following  short  table,  instead  of  com- 
puting these  corrections  by  the  formulae  of  §§  23  and  25. 
These  corrections  are  not  limited  to  meridian  altitudes, 
but  may  be  applied  to  any  observed  altitude  of  the  sun 
where  only  approximate  accuracy  is  required.  They 
correspond  to  an  average  condition  of  the  atmosphere 
represented  by  a  barometric  pressure  of  29.0  inches  and 
a  temperature  of  50°  Fahr. 

h1.         Ref.-Par. 
20°         -2'.4 


3° 

-1-5 

40 

—  I  .0 

50 

-0.7 

60 

-o-5 

70 

-0-3 

80 

—  O.I 

90 

o.o 

The  following  latitude  determination  was  made  by 
measuring  with  a  sextant  and  artificial  horizon  (§59) 
the  maximum  double  altitude  of  the  sun's  lower  edge 
(limb)  upon  a  date,  Dec.  19, 1898,  at  which  the  sun's  dec- 


PLATE  I. 


An  American  Engineer's  Transit.     Diameter  of  Horizontal  Circle  7  inches. 
Approximate  Cost  $350. 

[To  face  p.  62.] 


• 


ROUGH  DETERMINATIONS  OF  TIME,  ETC.  63 

lination,  as  furnished  by  the  almanac,  was,  d  =  —  23°26'.o< 
The  reduction  of  the  observation  is  as  follows : 


Sextant  Reading 

57°  44'  30" 

Instrumental  Corr. 

-  i    50 

Corr'd  Sextant 

57    42   40 

h' 

28    51.3 

Ref.-Par. 

-    1.6 

Semi-diameter 

+  16.3 

k 

29     6.0 

9o°  +  S 

66  34.0 

Latitude,  <£ 

37   28.0 

Make  a  determination  of  your  own  latitude  by  a  simi- 
lar method. 

30.  Time   and   Azimuth   from   an    Observed  Altitude. — 

If  the  latitude  be  thus  observed  at  noon,  time  may  be 
determined  with  a  sextant,  and  both  time  and  azimuth 
may  be  determined  with  a  theodolite,  by  measuring  an 
altitude  of  the  sun  when  it  is  at  some  considerable  dis- 
tance from  the  meridian,  e.g.,  when  its  azimuth  is  60° 
or  more.  Observations  of  this  kind,  if  made  for  the 
determination  of  time  only,  may  be  reduced  by  the 
method  developed  in  T§  36,  and  we  shall  here  treat  of 
observations  made  for  the  determination  of  both  time 
and  azimuth. 

There  should  be  at  least  two  such  observations  made, 
one  Circle  R.  and  the  other  Circle  L.,  in  order  to  eliminate 
instrumental  errors  (see  §  50).  Observe  the  edges  of 
the  sun,  not  its  centre,  and  correct  the  results  for  semi- 
diameter  (§  24);  but  if  the  instrument  is  provided  with 
stadia  threads,  this  correction  may  be  avoided  as  follows : 
Point  the  telescope  at  the  sun  so  that  the  two  horizontal 


64  FIELD  ASTRONOMY. 

threads  cut  off  equal  segments  from  the  upper  and  lower 
edges  of  the  sun,  and  by  turning  the  slow-motion  screw 
in  altitude,  keep  these  segments  of  equal  area  as  the  sun 
drifts  across  the  field  of  view,  until  it  reaches  a  position 
in  which  the  vertical  thread  bisects  each  segment. 
Record  this  time  to  the  nearest  second,  and  also  record 
the  readings  of  the  four  verniers  of  the  instrument. 
Before  reversing  the  instrument  to  obtain  the  second 
observation,  read  and  record  both  its  levels  (azimuth  , 
and  altitude  levels,  §§48  and  50),  and  after  reversing 
bring  the  bubbles  back,  by  means  of  the  levelling-screws, 
to  the  position  thus  recorded.  This  process  eliminates 
errors  of  level. 

The  better  class  of  engineer's  transits  are  usually 
provided  with  shade-glasses  to  moderate  the  intensity 
of  the  sun's  light  and  permit  it  to  be  viewed  through 
the  telescope.  But  these  glasses  are  by  no  means  neces- 
sary, since  an  image  of  the  sun  and  the  threads  of  the 
instrument  may  be  projected  upon  a  piece  of  cardboard 
and  be  there  seen  and  observed  quite  as  accurately  as 
in  the  telescope.  Pull  the  eyepiece  out,  away  from 
the  threads,  until  the  latter  can  no  longer  be  seen  dis- 
tinctly with  the  eye ;  then  allow  the  sun  to  shine  through 
the  telescope  upon  the  cardboard  held  behind  the  eye- 
piece, and  shift  the  cardboard  toward  and  from  the 
instrument  until  a  position  is  found  in  which  the  projected 
images  of  the  threads  appear  sharp  and  distinct.  Then 
turn  the  focussing-screw  until  the  edge  of  the  sun's 
image  also  appears  well  defined  and  the  projected  images 
will  be  ready  for  observation. 


ROUGH  DETERMINATIONS  OF  TIME,  ETC.  65 

Reduction  of  the  Observations. — We  shall  represent  by 
T  the  mean  of  the  two  recorded  times  of  observation 
of  the  sun,  by  Hf  the  mean  of  the  corresponding  read- 
ings of  the  horizontal  circle,  and  by  hr  the  measured 
altitude  furnished  by  the  two  observations.  The  cor- 
rections for  refraction  and  parallax  required  to  reduce 
the  instrumental  h'  to  the  sun's  true  altitude,  h,  may  be 
taken  from  the  table  in  §  29.  The  data  of  our  problem 
consist  of  the  three  sides  of  the  astronomical  triangle, 
which  we  shall  represent  by  the  letters  a,  b,  c,  and 
obtain  their  numerical  values  as  follows :  a  =  90°  —  d,  by 
interpolating  from  the  almanac  the  sun's  declination 
corresponding  to  the  time  T;  b  =  go°  —  h,  from  the 
observation;  and  £=-90°  — <#,  from  the  latitude,  which 
is  here  supposed  to  be  known.  To  these  data  we  ap- 
ply Equations  6  and  7  of  Chapter  I  and  determine 
the  three  angles  of  the  triangle,  viz.,  the  sun's  hour 
angle,  t,  its  azimuth  reckoned  from  the  north  point,  ANt 
and  the  third  angle  of  the  triangle,  q,  as  follows : 

Putting  5  =  i(a  +  6-K),  we  introduce  the  auxiliary 
quantity,  k,  defined  by  the  relation, 


,  _    I  _  sin  5 
\sin  s  -a   sin  s- 


_        _ 
sin  (s  -a)  sin  (s-b)  sin  (s-c)' 

and  find,  in  terms  of  it, 

cot  %AN=k  sin  (s—  a), 

=  &  sin  (<>-&),  (37) 

=k  sin  (s-c). 


66  FIELD  ASTRONOMY. 

For  an  A.M.  observation  use  the  —  ,  for  a  P.M.  observa- 
tion the  +  sign  for  k. 

The  angle  q,  for  which  we  have  no  direct  need,  is 
included  in  the  solution  for  the  sake  of  the  following 
'check'  which  it  furnishes  upon  the  numerical  com- 
putations :  Multiplying  together  the  three  equations  last 
given  and  replacing  k2  in  the  product  by  its  value  in 
terms  of  s,  we  obtain, 

cot  %An  cot  \t  cot  \q  =  k  sin  s,  (38) 

an  equation  that  must  be  satisfied,  within  one  or  two 
units  of  the  last  decimal  place,  by  the  numbers  obtained 
in  the  logarithmic  solution. 

The  student  should  not  fail  to  note  the  following 
relation  as  an  additional  check  to  be  applied  in  the  prog- 
ress of  the  numerical  work: 


b)  +  (s-c)=s.  (39) 

Transforming  the  hour  angle,  t,  into  mean  time  by 
means  of  the  equation  of  time,  E,  whose  value  corre- 
sponding to  the  instant  T  is  to  be  interpolated  from  the 
almanac,  we  obtain  as  the  chronometer  correction,  re- 
ferred to  local  mean  solar  time, 

JT  =  t  +  E-T.  (40) 

If  we  add  to  the  sun's  computed  azimuth,  AN,  the 
circle  reading,  Hr,  we  shall  obtain  the  index  correction 
of  the  circle,  i.e.,  the  vernier  reading  for  which  the  line 
of  sight  points  due  north,  and  the  azimuth  (from  north) 
of  any  terrestrial  point  may  then  be  found  by  subtract- 
ing from  the  vernier  reading  corresponding  to  it  the 


ROUGH  DETERMINATIONS  OF  TIME,  ETC. 


67 


index  correction  thus  found;  e.g.,  for  the  azimuth  of 
any  terrestrial  point  observed  in  connection  with  the 
sun  and  for  which  the  circle  reading  was  P',  we  have 

A=P'-(H'  +  AN).  (41) 

But  note  that  for  an  observation  made  before  noon  the 
negative  value  of  k  makes  ANa  negative  quantity.  The 
hour  angle,  t,  may  also  be  reckoned  as  negative  for  an 
A.M.  observation,  and  increased  by  24*. 

A  form  for  the  record  and  reduction  of  such  obser- 
vations is  shown  below  in  connection  with  a  set  of  obser- 
vations made  at  a  place  whose  latitude  and  longitude 
(from  Greenwich)  are  respectively  43°  4'. 6  and  5h  58m. 

ALTITUDES    OF    SUN'S    CENTRE. 

At  Station  A. — April  16,  1897. 
Engineer's  Transit,  B.     Watch  No.  6.     Observer,  C. 


Circle. 

Object. 

Watch. 

Vertical  Circle. 

Horizontal  Circle. 

Vcr.  A. 

Ver.  B. 

Ver.  A. 

Ver.B. 

R. 
R. 
L. 
L. 

Sta.  B 
Sun 
Sun 
Sta.  B 

h     in       s 

o      /       // 

/        // 

O             III 

187   54   10 
259    *4     o 
79  45      o 

7   54   10 

i      n 

54   10 
13  5° 
45     o 
54   10 

4    12    33 
4   15    18 

26   26      o 

25   57   20 

25    5° 
57     o 

d 

0 

Vert  Circ. 

Corr. 

h 

a 
b 
c 

2S 
S 

s—a 
s-b 
s—c 


10° 

28'.  o 

43 

4.6 

26 

"•S 

-1.8 

26 

9-7 

79 

32.0 

63 

50.3 

46 

55-4 

190 

17.7 

95 

8.8 

15 

36.8 

i8-5 

48 

13-4 

REDUCTION. 

sin  (s—  a) 

9.4300 

sin  (s—b) 

9-7I57 

sin  (s—c) 

9.8726 

Sum 

9.0183 

sin  5 

9.9982 

k2 

0.9799 

k 

0.4899 

cot  \A  N 

9.9199 

cot  |J 

o.  2056 

cot  %q 

0.3625 

Sum 

0.4880 

k  sin  s 

0.4881* 

AN 

H' 

North 

t  (arc) 
t  (time; 

t+E 

T 

AT 

P' 

A 


100"  30'. 4 

79     29.5 

J79     59-9 

63     50.0 
4h  i5m  20" 

-  o  25 
4  14  55 
4  13  55 

+    i       o 

187°  54'. 2 
7    54-3 


68  FIELD  ASTRONOMY. 

The  true  azimuth  of  Station  B  was  known  to  be 
187°  54'  I0">  and  a  comparison  of  the  watch  with  standard 
time  furnished  as  the  true  value  of  AT,  +57  seconds. 
The  differences  between  these"  results  and  those  found 
in  the  preceding  solution  furnish  a  fair  idea  of  the  pre- 
cision to  be  expected  in  such  work. 

31.  Time  by  Meridian  Transits. — If  astronomical  ob- 
servations are  to  be  made  for  any  considerable  length 
of  time  at  a  given  station,  as  at  a  university,  it  will  be 
convenient  for  many  purposes  to  determine  the  azimuth 
of  a  permanently  marked  line,  at  one  end  of  which  an 
instrument  can  be  set  up  and  oriented.  If  a  theodolite 
be  thus  mounted  and  its  line  of  sight  brought  into  the 
meridian,  a  time  determination  may  be  very  simply  made 
by  observing  the  chronometer  time  of  transit  of  the 
sun's  .preceding  and  following  limbs  past  the  vertical 
thread  of  the  instrument.  Since  the  thread  is  by  sup- 
position in  the  meridian,  the  hour  angle  of  the  sun  at  the 
mean  of  the  observed  times,  T,  is  zero  and  we  have 

JT  =  a-T  (Sidereal), 
or  JT  =  E—T  (Mean  solar).  (42) 

If  the  azimuth  of  the  line  is  well  determined,  this 
method  may  rank  as  an  approximate  rather  than  a 
rough  determination,  since  under  ordinary  circumstances 
there  must  be  an  error  of  nearly  2'  in  the  orientation 
of  the  instrument,  to  produce  an  error  of  6s  in  the  chro- 
nometer correction.  In  any  case  the  instrument  must 
be  carefully  levelled,  particularly  in  the  east  and  west 
direction,  and  in  the  following  example  the  readings 


ROUGH  DETERMINATIONS  OF  TIME,  ETC. 


of  the  striding  level  are  employed  as  a  control  upon  this 
adjustment. 

Observe  the  slight  variation  of  method  here  intro- 
duced in  order  to  obtain  in  place  of  a  single  observation 
two  observations,  one  Circle  R.,  and  one  Circle  L. 

TRANSITS    OF    SUN   FOR   TIME    DETERMINATION. 
At  Station  A.     April  17,  1897. 

Theodolite,  F.     Watch  No.  6.     Observer,  G. 
Instrument  oriented  on  Station  B. 


Circle. 

Ver.  A. 

Limb. 

Watch. 

Striding 

Reduction. 

level. 

0               / 

h.     m.     s. 

T  =  iih  57ra  37s 

R. 
L. 

359  30 
180  30 

Pr. 
Fol. 

I*    55    34 
ii    59  40 

1.4    12.6 
14.0      0.5 

i2b+£  =  n   59     23 

By  a  comparison  with  standard  time  the  true  AT  referred  to  the 
local  meridian  was  found  to  be  +  im  47'. 

The  telescope  of  an  engineer's  transit  is  usually 
capable  of  showing  a  first-magnitude  star  by  daylight 
whenever  the  sky  is  clear  and  blue,  and  such  a  star  is 
equally  available  with  the  sun  for  a  determination  of 
either  latitude  or  time.  In  the  spring  and  summer 
Sirius,  by  reason  of  its  great  brilliancy,  is  a  peculiarly 
favorable  object  for  such  observations  (see  Table  V 
for  the  approximate  right  ascension  and  declination 
of  this  and  other  stars).  Even  the  brightest  of  these 
stars  is  not  a  conspicuous  object  by  daylight,  and  is  most 
readily  found  by  placing  the  telescope  in  the  meridian 
and  at  the  proper  zenith  distance,  z  =  <t>—d,  and  await- 
ing its  arrival  in  the  field  of  view.  A  very  slight  error 
of  focus  in  the  telescope  will  render  the  star  invisible, 
and  this  adjustment  should  therefore  be  carefully  made 


70  FIELD  ASTRONOMY. 

upon  a  distant  terrestrial  object  before  setting  the  tele- 
scope for  the  star. 

32.  Orientation  by  Polaris. — If  a  rough  determination 
of  time,  latitude,  or  azimuth  is-; to  be  made  by  night,  or  if 
a  theodolite  is  to  be  oriented  as  a  preparation  for  other 
work,  observations  of  the  Pole  Star  by  the  following 
method  will  be  found  especially  convenient,  since  no 
almanac  is  required  and  no  instrumental  equipment 
other  than  an  engineer's  transit  and  a  watch  approxi- 
mately regulated  to  local  mean  solar  time. 

If  Polaris  were  exactly  at  the  pole  of  the  heavens, 
the  instrument  might  be  oriented  by  pointing  directly 
upon  the  star,  and  setting  the  verniers  to  read  180°, 
and  simultaneously  the  latitude  might  be  determined 
by  measuring  the  star's  altitude,  since  in  this  case,  <f>  =  h. 
As  Polaris  is  actually  more  than  a  degree  distant  from 
the  pole,  this  ideal  method  is  inapplicable,  but  the  prin- 
ciples upon  which  it  is  based  may  be  applied  by  means 
of  the  tables  at  the  end  of  this  book,  which  furnish 
directly,  for  the  year  1900  and  for  the  latitude  40°,  the 
amounts,  a  and  6,  by  which  the  azimuth  and  altitude 
of  Polaris  differ  at  any  moment  from  the  corresponding 
coordinates  of  the  pole.  The  argument  of  the  first  table, 
/,  is  the  star's  hour  angle,  and  its  value  at  any  given 
moment  may  be  determined  from  an  ordinary  watch 
as  follows:  If  AT  represents  the  correction  required  to 
reduce  the  watch  time,  T,  to  local  mean  solar  time,  we 
shall  have  as  the  hour  angle  of  the  mean  sun  at  the  in- 
stant r, 


ROUGH  DETERMINATIONS  OF  TIME,  ETC.  71 

Once  in  each  year  Polaris  and'  the  mean  sun  have  the 
same  right  ascension  and  therefore  the  same  hour  angle; 
and  if  we  represent  this  date  by  E,  we  shall  have  for  the 
difference  A'  of  their  hour  angles  on  any  other  date,  D, 
an  expression  of  the  form 

A'  =  Star-  Sun  =  C(D-E), 

where  C  is  the  daily  increase  of  the  mean  sun's  right 
ascension  over  that  of  the  star,  and  the  interval,  D  —  E, 
is  to  be  expressed  in  days.  In  minutes  of  time, 
C  =  4(i  — TV)  and,  therefore, 

A'  =  4(D  -  E)  { i  -  TV } ,  (minutes.)  (43) 

For  the  date  D  we  have,  therefore,  as  the  expression 
of  the  Pole  Star's  hour  angle, 

*  =  T  +  jr  +  J'.  (44) 

The  correction  to  the  watch,  JT,  need  be  only  roughly 
known,  e.g.,  within  two  or  three  minutes,  or  even  more; 
and,  correspondingly,  the  value  of  the  last  term  in  this 
expression  need  be  computed  only  to  the  nearest  minute. 
Table  IV  gives,  to  the  nearest  tenth  of  a  day,  the  value 
of  E  for  each  year  from  1900  to  1930,  expressed  in  the 
mean  solar  time  of  the  meridian  90°  west  of  Greenwich. 
The  date  D  is  to  be  similarly  expressed,  and  it  must  be 
remembered  that  in  astronomical  practice  the  day  begins 
at  noon,  so  that,  for  example,  an  observation  made  at 
5  A.M.  on  May  10  has  for  the  corresponding  value  of  D, 
May  9.7.  With  the  value  of  t  furnished  by  Equation  44 


W  FIELD  ASTRONOMY. 

we  may  interpolate  from  Table  I  the  quantities  a  and  6 
corresponding  to  the  position  of  Polaris  as  seen  in  the 
year  1900  by  an  observer  in  4.0°  north  latitude.  The 
proper  algebraic  signs  to  be  used  with  a  and  b  are  printed 
in  Table  I,  preceding  or  following  the  numbers  according 
as  the  argument  t  is  found  in  the  left-  or  right-hand 
column  of  the  table,  the  plus  sign  indicating  that  the 
star  has  a  greater  azimuth  or  altitude  than  that  of  the 
pole. 

Since  both  a  and  b  depend  upon  the  star's  distance 
from  the  pole,  which  varies  from  year  to  year,  and  since 
a  is  also  a  function  of  the  observer's  latitude,  these 
interpolated  ,  quantities  will  in  general  require  some 
correction  in  order  to  give  the  star's  real  position  with 
respect  to  the  pole.  We  therefore  write  as  the  coor- 
dinates of  Polaris, 


A  =  iSo°  +  F1a,         h  =  </>  +  F2b,  (45) 

where  the  coefficients,  F1  and  F2,  are  factors  required 
to  transform  the  tabular  a  and  b  into  the  required  quan- 
tities that  fit  the  time  and  place  of  observation.  The 
numerical  value  of  F2  may  be  interpolated  from  Table  III, 
with  the  year  in  which  the  observation  was  made,  as 
the  argument,  while  Fl  must  be  interpolated  from 
Table  II  (double  entry),  with  the  year  and  the  observer's 
approximate  latitude  as  arguments. 

Note  that  for  a  given  place  E,  Fv  and  F2  are  con- 
stant for  a  year,  and  when  once  interpolated  should 
be  written  down  and  preserved  for  future  use. 

To  illustrate  the  use  of  the  tables  we  take  the  fol- 


ROUGH  DETERMINATIONS  OF  TIME,  ETC. 


73 


lowing  observations  made  in  latitude  approximately 
43°,  with  a  carefully  adjusted  engineer's  transit  and  a 
watch  supposed  to  be  three  minutes  slow  of  local  mean 
solar  time. 

Saturday,  April  26, 1902. 
At  Station  A.     Theodolite,  F.     Observer,  C. 


Object. 

Circle. 

Watch. 

Az.  Circle. 

Alt.  Circle. 

Polaris      

L. 

Qh   20m  —  s 

179°  17' 

41°    co' 

B  Virginis  

L. 

Q       2Z       4< 

o      o 

4Q      14 

Mark      

L. 

9     28     — 

308     I 

—  o     43 

Mark        

R. 

o      20     — 

308      o 

—  o     42 

During  the  ten  minutes  preceding  gh  2om  the  coor- 
dinates of  Polaris,  h  and  A,  were  computed  for  the  time 
9h  20ra  (Equation  45 ;  mental  arithmetic,  without  writ- 
ing down  a  figure  except  the  interpolated  values  of 
E,  Fv  F2,  which  are  to  be  preserved  for  future  use). 
The  course  of  the  computation  was  as  follows: 


D 
E 
(D-E) 
A' 
T  +  AT 
t 

April  26.4 
April  13.2 
+  53 

h+52 

gh     2^m 

10     15 

PI 

a 

A 

1.05 


-43 
1 80°  o' 
179  17 


?• 

F. 


I  .00 

-7°65' 

43     o 
4i   55 


After  completing  this  computation  the  telescope  was 
set  to  the  computed  altitude  41°  55',  and  the  A  Vernier 
of  the  horizontal  circle  was  set  to  read  179°  if.  Then, 
turning  the  instrument  about  the  lower  motion,  Polaris 
was  found,  and  at  gh  2om  by  the  watch  the  star  was 
brought  behind  the  intersection  of  the  cross-wires,  with- 
out disturbing  the  reading  of  the  horizontal  circle,  and 
the  vertical  circle  was  read  and  recorded  as  shown  above. 


74  FIELD  ASTRONOMY. 

The  instrument  being  now  oriented,  was  turned 
into  the  meridian  by  making  Vernier  A  read  o°  o',  the 
telescope  was  set  to  the  computed  altitude  of  the  star 
/?Virginis  (Table  V), 

/*  =  90°-0+d  =  49°  19', 

and  the  time  of  its  transit  behind  the  vertical  thread 
observed,  as  in  §  31,  at  the  recorded  time,  9h  2$m  45s. 

Readings  to  a  distant  mark,  an  electric  light,  were 
then  taken  to  determine  its  azimuth,  and  this  observa- 
tion was  repeated  in  the  reversed  position  of  the  instru- 
ment as  a  check  upon  errors  of  adjustment.  The  close 
agreement  of  these  readings,  Circle  L.  and  Circle  R.,  shows 
the  adjustment  to  be  satisfactory,  and  we  have  imme- 
diately, as  the  true  azimuth  of  the  mark,  the  circle  read- 
ing 308°  i'. 

Since  F2b=h—(/>,  we  obtain  from  the  preliminary 
computation  and  the  reading  of  the  vertical  circle, 

0  =  4i°  59'  + 1°  5' =43°  4', 

which  is  within  a  minute  of  the  known  latitude  of  the 
instrument. 

For  a  time  determination  we  obtain  from  the  almanac 
the  right  ascension  of  /?Virginis,  a  =  nh  45 m  38s,  and 
subtracting  from  this  the  observed  time,  9h  25™  45s, 
we  find 

AT  =  -f  2h  i9m  53s.     (Referred  to  sidereal  time.) 

A  comparison  of  the  watch  with  a  standard  sidereal 
clock  furnished  as  the  true  value  oi  AT,  +2hi9m55s. 


ROUGH  DETERMINATIONS  OF  TIME,  ETC.  75 

The  entire  determination  of  azimuth,  latitude,  and 
time  thus  made,  occupied  less  than  thirty  minutes, 
including  both  computation  and  observation.  The  ac- 
curacy of  the  results  obtained  is  fairly  typical  of  what 
may  be  expected  from  the  method  when  instrument 
and  tables  are  carefully  used. 

If  the  assumed  correction  of  the  watch,  JT=  +  3m,  were  wide  of 
the  truth,  serious  error  might  be  introduced  into  the  results,  and  we 
have  now  to  learn  whether  the  assumed  JT  was  in  fact  seriously  wrong. 
Since  T  +  AT=a  +  t,  we  find  for  the  instant  of  orientation,  using  for 
the  right  ascension  of  Polaris  the  value  of  its  a  given  in  Table  III, 


t  =9h  2om  +  2h  19™  53"  —  ih  24m  =ioh  i6m, 

which  is  a  sufficiently  close  agreement  with  the  assumed  hour  angle, 
ioh  15™,  to  confirm  the  assumed  AT=  +  3™. 

The  right  ascension  of  the  time  star,  in  this  case  /?  Virginis,  should 
be  taken  from  the  almanac  whenever  one  is  available,  but  in  the 
absence  of  an  almanac  the  method  above  outlined  may  still  be  applied 
through  the  use  of  these  tables,  without  overstepping  the  limits  of 
error  adopted  for  a  rough  determination.  Table  V  contains  a  list 
of  time  stars  suitable  for  observation  with  an  engineer's  transit,  and 
gives  their  declinations  to  the  nearest  minute  and  their  right  ascen- 
sions to  the  nearest  second  (neglecting  the  nutation),  for  the  year 
1900  and  the  date  contained  in  the  last  column  of  the  table.  The 
given  dates  are  those  at  which  the  stars  come  to  the  meridian  at 
8  P.M.,  local  mean  solar  time,  and  it  is  presumed  that  this  will  be,  on 
the  whole,  the  most  convenient  hour  for  observation.  But  any  star 
that  crosses  the  meridian  before  midnight  may  be  observed  and  its 
right  ascension  for  the  given  date  obtained  from  the  table  within  one 
or  two  seconds  by  adding  to  the  a  there  given,  the  amount  of  the 
annual  variation  (Ann.  Var.  in  the  fourth  column)  multiplied  by  the 
number  of  years  that  have  elapsed  since  1900.  Applying  Table  V 
to  the  preceding  example,  we  obtain  for  the  right  ascension  of  /?  Vir- 
ginis in  1902,  nh  45m  3is-f  2  X3-i  =nh  45m  37s,  agreeing  within  one 
second  with  that  furnished  by  the  almanac.  The  declinations  of 
these  stars  are  also  subject  to  an  annual  variation  which,  for  the  present 
purpose,  may  be  ignored. 

As  an  aid  to  identifying  the  particular  stars  convenient  for  obser- 
vation at  a  given  time,  we  note  that  having  computed  the  hour  angle 
of  Polaris,  f,  corresponding  to  9h  20™  by  the  watch,  we  may  obtain 


76  FIELD  ASTRONOMY. 

the  corresponding  sidereal  time  by  adding  to  t  the  right  ascension  of 
Polaris  as  shown  in  Table  III,  a  +  t  =  d.  We  find  thus  nh  39'"  as  the 
sidereal  time  corresponding  to  9b  20™  by  the  watch,  and  Table  V 
shows  by  its  column  of  right  ascensions  that  the  first  star  coming  to 
the  meridian  after  the  orientation  at  9h  20™  was  /?  Virginis,  which 
was  therefore  chosen  as  the  star  to  be  observed.  The  time  9h  2om 
was,  in  fact,  selected  with  reference  to  having  a  suitable  time  star 
available  immediately  after  the  orientation. 

33.  Mathematical   Theory  of  the   Quantities    a,    b,    Flt 

and  F2.  —  If  in  Equations  14,  for  the  transformation  of 
coordinates,  we  replace  d  by  its  equivalent  90°  —  p,  where 
p  is  the  distance  of  Polaris  from  the  pole,  we  may  sub- 
stitute for  the  resulting  sin  p  and  cos  p  their  values, 

sin  p  =  p,          cos  p  =  i  —  %p2, 

with  a  similar  substitution  for  sin  A  and  cos  A  ,  and 
obtain,  in  place  of  the  rigorous  transformation  formulae, 
approximate  ones  more  convenient  and  sufficiently 
accurate  for  our  purpose.  The  maximum  quantity 
neglected  in  this  substitution  is  of  the  order  p3,  which, 
in  the  case  of  Polaris,  amounts  to  less  than  2". 

Leaving  to  the  reader  the  details  of  this  substitution, 
we  write  down  as  the  resulting  development  of  A  and  h, 
correct  to  terms  of  the  order  p2,  inclusive, 


A  —  1  80°  =  —  £sec0sin2  —  J(£sec  0)  2sin  0sin  2t  +  etc., 

14-6) 

h-    0    = 


where  the  last  term  in  the  expression  for  h  represents 
approximately  the  effect  of  refraction  in  increasing  the 
star's  apparent  altitude.  With  an  assumed  latitude, 
0  =  40°,  an  assumed  value  of  p  corresponding  to  the 


ROUGH  DETERMINATIONS  OF  TIME,  ETC.  77 

epoch  1900  and  represented  by  the  symbol  pQ,  and  with 
a  value  of  the  coefficient  R  corresponding  to  an  average 
condition  of  the  atmosphere  (Thermometer,  50°  Fahr., 
Barometer,  29.00  inches),  the  second  members  of  these 
equations  have  been  tabulated  as  the  a  and  b  of  Table  I. 

It  is  evident  from  an  inspection  of  the  equations  that 
the  factors  required  to  change  the  tabular  a  and  b  into 
the  coordinates  corresponding  to  a  different  year  and 
place  (different  values  of  p  and  0)  are  approximately 

p  p  sec  0 

F*  =  PQ'     F'=^~0T/F2'  (47) 

where  /  represents  that  part  of  Fl  that  depends  upon  the 
latitude.  The  values  of  Fl  and  F2  contained  in  Tables  II 
and  III  are  derived  from  these  expressions,  but  since 
these  are  only  approximate  and  neglect  some  small  fac- 
tors of  the  problem,  a  certain  amount  of  additional  error 
is  thereby  introduced,  so  that  the  resulting  coordinates 
of  Polaris  in  some  cases  may  be  more  than  a  minute  of 
arc  in  error;  e.g.,  the  aberration  of  light  and  the  nutation 
are  entirely  neglected  in  this  analysis,  as  is  also  the  vary- 
ing amount  of  the  refraction  in  different  latitudes,  etc. 

In  considerable  part  these  influences  may  be  taken  into  account 
and  the  precision  of  the  results  somewhat  increased  by  using  values 
of  the  factors  Flt  F2,  obtained  from  the  following  supplementary  tables 
instead  of  the  values  contained  in  Tables  II  and  III.  The  argument 
of  Table  B  is  to  be  determined  by  the  season  of  the  year  at  which  the 
observation  is  made  and  may  be  either  the  year  itself  or  the  preceding 
or  following  year,  as  shown  below: 

As  Argument  for  Table  B,  Use 

The  Preceding  Year  in       March,  April,  May,  June,  July. 

The  Given  Year  in  January,  February,  August,  September. 

The  Following  Year  in       October,  November,  December. 


78  FIELD  ASTRONOMY. 

The  error  of  refraction  is  of  consequence  only  in  latitude  determina- 
tions, and  to  correct  its  effect  we  have  to  subtract  from  the  observed  lati- 
tude the  quantity  A<f>  given,  in  minutes  of  arc,  in  Table  A. 

TABLE  A. '  TABLE  B. 

0.  Log/.  J<f>.  Year.  LogF2. 

10°          9.891  4  1900          o.ooo 

15  9.899  2  1905          9.991 

20  9.9H  I  1910  9.881 

25        9.927  I  1915       9.973 

30  9-947  o  1920         9.963 

35  9.971  o  1925         9.952 

40  o.ooo     2Q  o  J93o          9.942 

45  0.035     3S  °  J935          9-931 

56          0.076     4I  o  1940         9.921 


Let  the  student  derive  from  these  tables  the  value  of  Fj  and  F2 
corresponding  to  the  observation  reduced  in  §  32  and  compare  them 
with  the  values  there  used. 

3 3 a.  Artificial  Illumination. — For  the  observation  of 
stars  by  night  there  must  be  provided  some  artificial 
illumination  for  the  telescope  as  well  as  for  the  verniers, 
since,  otherwise,  the  threads  that  determine  the  line  of 
sight  (cross- wires)  will  be  invisible.  For  this  purpose 
there  are  several  mechanical  devices  by  which  the  light 
from  a  bull's-eye  lantern  or  electric  hand-lamp  may  be 
reflected  into  the  field  of  view  of  the  telescope,  but  for 
a  small  instrument  none  of  these  possess  any  marked 
advantage  over  a  bit  of  candle-grease,  dropped  in  the 
liquid  state  and  allowed  to  cool  upon  the  center  of  the 
objective.  Pare  it  down  thin  with  a  penknife  and  throw 
the  light  upon  it  along  a  line  but  little  inclined  to  the 
axis  of  the  telescope.  The  effect  of  the  grease  upon  the 
optical  performance  of  the  telescope  is  quite  insensible. 


CHAPTER  VI. 
APPROXIMATE   DETERMINATIONS. 

34.  Latitude  by  Circum-meridian  Altitudes.  —  An  ob- 
vious method  of  refining  upon  the  rough  determination 
of  a  latitude  from  a  single  observation  of  the  meridian 
altitude  of  the  sun  or  a  star  (as  in  §  29)  is  to  measure  a 
series  of  altitudes  during  the  few  minutes  preceding  and 
following  the  maximum  h,  and  to  derive  from  all  these 
observations,  which  are  called  circum-meridian  altitudes, 
a  better  value  of  the  meridian  altitude  than  a  single 
measurement  can  be  expected  to  furnish.  t  Each  meas- 
ured altitude  will  usually  differ  from  the  maximum 
altitude  by  an  amount  called  the  reduction  to  the  merid- 
ian, and  this  reduction  may  be  accurately  computed 
if  either  the  hour  angle  or  the  azimuth  of  the  star  at  the 
time  of  observation  is  known. 

If  the  observations  are  made  with  a  sextant,  the  hour 
angle  will  be  most  convenient  for  the  reduction,  and 
the  time  of  each  observation  should  therefore  be  noted, 
to  the  nearest  second,  by  the  use  of  some  watch  or  other 
timepiece.  To  obtain  a  convenient  method  of  reduc- 
tion for  the  observations  we  put  /  =  o  in  the  equation, 

sin  h  =  sin  0  sin  d  +  cos  0  cos  d  cos  t,  (48) 

79 


80  FIELD  ASTRONOMY. 

and  obtain  for  the  maximum  altitude 

sin  h0  =  sin  0  sin  d  +  cos  0  cos  d.  (49) 

Since  in  the  cases  here  considered  the  hour  angles  are 
not  to  exceed  iom  or  15™,  we  may  put  cost  =  i  —  J£2, 
and  subtracting  the  first  of  the  preceding  equations 
from  the  second  obtain, 


2  sin  %(h0  —  h)  cos  %(h0  +  k)  =\  cos  0  cos  d  .  t2,     (50) 
which  is  approximately  equivalent  to, 

,  _  cos  0  cos  d         t2 
^~  ~  "7' 


This  is  the  equation  of  a  parabola  having  h0  for  its  maxi- 
mum ordinate  and  h  and  t  for  rectangular  coordinates; 
and  we  may  infer  from  it  that  if  the  sextant  readings 
be  plotted  as  ordinates  upon  cross-section  paper  with 
the  observed  times  as  abscissas,  the  resulting  curve  will 
be  a  parabola  whose  maximum  ordinate  will  be  the 
sextant  reading  corresponding  to  the  maximum  altitude 
of  the  body  observed. 

This  maximum  ordinate  may  be  read  directly  from 
the  curve,  or  it  may  be  derived  with  greater  precision 
by  means  of  the  theorem  that  the  area  included  between 
a  parabola  and  any  chord  perpendicular  to  its  axis, 
equals  two  thirds  of  the  length  of  this  chord  multiplied 
by  its  distance  from  the  vertex,  A  =  %xy.  The  inter- 
cept of  the  plotted  curve  upon  the  axis  of  x,  or  upon 
any  line  parallel  to  this  axis,  is  such  a  chord,  whose 


APPROXIMATE  DETERMINATIONS. 


81 


length  may  be  directly  measured,  and  the  distance  of  the 
vertex  from  this  axis  is  the  quantity  sought.  If,  there- 
fore, the  length  of  the  intercept,  #,  and  the  area  of  the 
corresponding  part  of  the  curve,  A,  be  directly  measured, 
we  have  at  once, 


2  OC 

Friday  May,  4  1897. 

Sextant  No.  5096.     Index  corr.  —3'  34".     Observer,  C. 
Barometer  29.10.     Thermometer  69°  Fahr. 

Horizon  Roof  Direct.  Horizon  Roof  Reversed. 


(52) 


Limb. 

Watch. 

Sextant. 

h.    m.      s. 

0               /               // 

L. 

II    44   33 

125   36   10 

L. 

46    15 

125   39     o 

u. 

50    27 

126  47   45 

u. 

51    23 

126  48   20 

L. 

53   i5 

125  46   10 

Limb. 

Watch. 

Sextant. 

h.    m.     s. 

0                 /              // 

U. 

ii   55  45 

126  50     5 

u. 

58     3 

126  49  45 

u. 

12        0       8 

126  48     o 

L. 

5  43 

125   34  40 

L. 

6  30 

125   32   30 

The  author  finds  from  the  area  of  the  curve  the  following  value  of 
y0,  from  which  the  latitude  is  derived  as  below. 


T   j 

Index  corr. 

h' 
Ref.-Par. 


126°  1 8'  36" 

-3  34 

63       7  3i 

-o  24 


63°    7'    7' 
26    52   53 
16    ii  41 
43      4   34 


The  axis  of  the  plotted  parabola  intersects  the  time  scale  at  iih 
55.0"',  and  comparing  this  number  with  the  local  mean  time  of  appar- 
ent noon,  i2h+£  =  nh  56m.6,  we  obtain  as  the  correction  required 
to  reduce  the  watch  to  local  mean  solar  time  JT=+i.6m.  A 
value  of  AT  thus  determined  may  easily  be  in  error  by  20  or  30  seconds. 

Among  the  advantages  of  this  mode  of  treatment  of 
the  data  may  be  noted  that  each  observation  contributes 
its  appropriate  share  toward  determining  the  maximum 
altitude  of  the  body,  and  that  no  knowledge  of  the  error 
of  the  timepiece  is  required.  In  fact  the  correction  JT 


82  FIELD  ASTRONOMY. 

may  be  approximately  determined  from  the  curve,  by 
noting,  as  the  chronometer  time  of  apparent  noon,  the 
point  at  which  the  axis  of  the  parabola  intersects  the 
axis  of  x. 

Let  the  student  plot  the  preceding  observations  made 
upon  the  sun's  upper  and  lower  limb,  and  derive  from 
the  area  of  the  curve  the  sextant  reading  corresponding 
to  the  sun's  meridian  altitude.  Before  plotting,  each 
sextant  reading,  double  altitude,  must  be  corrected  by 
twice  the  sun's  semi-diameter,  interpolated  from  the 
almanac  for  the  date  of  observation,  i.e.,  ±31' 47",  in 
order  to  obtain  the  corresponding  reading  to  the  sun's 
centre. 

For  an  approximate  method  of  dete'rmining  latitude 
from  altitudes  of  Polaris  the  student  may  consult  the 
American  Ephemeris,  Table  IV,  and  explanations  at 
the  end  of  the  volume. 

35.  Reduction  to  the  Meridian.  —  If  circum-meridian 
altitudes  are  to  be  measured  with  a  theodolite,  it  will 
usually  be  convenient  to  orient  the  instrument  and  deter- 
mine from  a  reading  of  the  horizontal  circle  the  azi- 
muth corresponding  to  each  observation.  A  graphical 
solution  may  then  be  made  precisely  as  in  the  case  of 
the  observed  times  treated  in  the  preceding  section,  or 
we  may  derive  from  Equations  15, 

sin  d  =  sin  0  sin  h  —  cos  0  cos  h  cos  A,  (53) 

and  from  this,  by  the  method  of  §  34,  we  find  the  relation, 

A2 
hQ—h=  cos  0  cos  h0  sec  d  .  — h  etc.  (54) 


APPROXIMATE  DETERMINATIONS.  83 

Through  this  equation  and  the  known  values  of  A,  com- 
pute for  each  observed  altitude  its  own  reduction  to 
the  meridian. 

The  quantities  h0—  h  and  A  are  here  supposed  to  be 
expressed  in  radians,  but  in  practice  it  is  convenient  to 
express  the  azimuth  in  minutes  and  the  reduction  to 
the  meridian  in  seconds  of  arc.  Representing  the  azimuth 
when  so  expressed  by  a',  we  make  in  Equation  54  the 
following  substitutions  : 


A  (radians)  =a'  .  ,  (A.-A)  (radians) 


and  uniting  into  one,  all  the  numerical  factors  that  are 
found  in  the  equation  as  thus  altered,  and  introducing 
the  symbol  /  as  an  abbreviation  for  the  product  of  all 
factors  not  containing  a',  we  obtain, 

/  =  [7-9407]  cos  0  cos  /*0  seed, 
(hQ-h)"=f(a'r. 

The  accents,  ',  ",  denote  that  the  marked  terms  are 
expressed  in  minutes  and  seconds  respectively.  Use 
an  estimated,  approximate,  value  of  h0  for  the  compu- 
tation of  /. 

The  preceding  results  cannot  be  directly  applied  to 
a  star  north  of  the  zenith,  since  for  such  a  star  the  azimuth, 
A,  is  a  large  quantity;  but  if  the  azimuth  be  reckoned 
from  the  north  point  instead  of  from  the  south,  i.e.,  if 
we  put  a'  =  180°-^,  we  may  derive  formulas  identical 
with  the  above,  which  therefore  apply  to  this  case  when 
a'  is  defined  as  the  supplement  of  the  azimuth.  For  a 


84  FIELD  ASTRONOMY. 

star  at  lower  culmination,  i.e.,  on  the  meridian  below 
the  pole,  the  altitude  is  a  minimum  instead  of  a  maxi- 
mum, and  the  reduction  to  the  meridian  must  therefore 
be  given  the  negative  sign.;"  Note  that  this  can  be 
accomplished  in  Equation  55  by  considering  d  to  repre- 
sent the  supplement  of  the  star's  declination  instead 
of  the  declination  itself.  These  formulae  for  reduction 
to  the  meridian  should  not  be  applied  in  the  case  of  stars 
whose  hour  angles  exceed  iom  or  i5m.  For  an  applica- 
tion of  the  formulae  see  §  73. 

36.  Time  from  Altitudes  near  the  Prime  Vertical. — 
With  a  sextant  an  approximate  determination  of  time 
is  best  made  by  measuring  a  series  of  altitudes  of  the 
sun  or  a  star  when  the  body  is,  as  near  as  may  be,  due 
east  or  west,  noting  the  chronometer  time,  T,  of  each 
observation. 

The  formulae  for  the  transformation  of  coordinates 
furnish  for  each  such  observation  the  equation, 

sin  h  =  sin  0  sin  d  +  cos  0  cos  d  cos  t, 

•* 
which  is  readily  transformed  into, 

cos  t  =  sec  0  sec  d  sin  h  —  tan  0  tan  d,  (56) 

and  by  means  of  this  equation  the  hour  angle  corre- 
sponding to  each  observed  time  may  be  derived.  The 
chronometer  correction  will  then  be  furnished  by  one 
of  the  following  equations: 

For  the  Sun,  4T=E  +  t-T,  Local  Mean  Solar  Time. 

For  a  Star,    AT  =  a  +  t-T,  Local  Sidereal  Time.          57' 


APPROXIMATE  DETERMINATIONS. 


85 


The  symbol  E  denotes  the  equation  of  time.  Its  numeri- 
cal value  is  most  conveniently  derived  from  the  Solar 
Ephemeris,  p.  400  of  the  almanac. 

DOUBLE    ALTITUDES    OF    ARCTURUS,    NEAR    EASTERN 
PRIME    VERTICAL. 

Wednesday,  March  29,  1899. 

Sextant,  Cameron.     Chronometer,  B.     Observer,  C. 
Index  Corr.  +18'  37".     Barom.  28.81.     Therm.  + 1 9°  Fahr. 


iextant.                 Chronometer. 

a  +  t. 

AT. 

0 

' 

h.    a 

I.      S. 

h.    m.    s. 

s. 

54 

3° 

9    30 

5-5 

9  29 

c 

,6 

-59 

9 

55 

o 

31 

29 

3° 

27. 

9 

61 

,i 

55 

3° 

32 

50 

31 

5° 

.  2 

59 

,8 

Horizon  roof  reversed. 


56  o 

56  30 

57  o 


34  14 

35  36 

36  58-5 


33  I2-5 

34  34-8 

35  57-1 
MeanJT.. 


61.5 

61.2 

61.4 

-60.8 


<j) 

43     4  37 

Corr'd  Sext. 

54  48  37 

57   18  37 

6 

19  42     9 

App't  h 

27   24   18 

28  39   19 

sec  <£ 

0.13642 

Refraction 

i   55 

i   49 

sec  6 

0.02620 

h 

27     22     23 

28  37   30 

tan  0 

9  .97082 

sin  h 

9.66255 

9  .  68040 

tan  6 

9-55401 

sec  <f>  sec  d  sin  h 

9.82517 

9.84302 

sec  0  sec  S 

o.  16262 

tan  (j>  tan  d 

9  .  52483 

9-52483 

Subtract 

9  .99862 

0.03367 

h.    m.       s. 

cos  t 

9-52345 

9-55850 

a 

14    II    6.1 

-t 

4  42   0.5 

4  35      9-° 

a+t 

9   29   5.6 

9  35   57-i 

Adopted  JT=  — 6o8. 8  (Local  sidereal). 

In  place  of.  the  laborious  process  of  separately  reduc- 
ing each  observed  altitude  we  may  usually  treat  the 
mean  of  the  sextant  readings  and  the  mean  of  the  ob- 
served times  as  if  they  constituted  a  single  observation. 
When  the  observed  body  is  near  the  prime  vertical  the 
time  interval  covered  by  a  set  of  observations  which  it 
is  purposed  to  unite  into  a  mean  result  may  extend  to 
15  or  20  minutes  without  sensible  error,  but  the  error 


86  FIELD  ASTRONOMY. 

of  the  process  increases  rapidly  with  increasing  distance 
from  the  prime  vertical,  and  the  time  interval  mtist  be 
correspondingly  diminished. 

In  the  preceding  example  of  a  time  determination 
from  sextant  altitudes,  the  sextant  was  set  accurately  to 
a  set  of  readings  differing  by  a  uniform  interval  of  30', 
and  the  times  noted  at  which  the  observed  body  came 
to  the  corresponding  altitudes.  In  the  reduction  ad- 
vantage is  taken  of  this  circumstance  by  computing  the 
value  of  a  +  t  for  the  first  and  last  observations  only, 
and  interpolating  the  intermediate  values.  Observe 
that  the  columns  a  +  t  and  AT,  although  placed  near  the 
beginning  of  the  reduction,  are  really  the  last  to  be  filled 
out. 

37.  Azimuth  Observations  at  Elongation.  —  An  excel- 
lent approximate  determination  of  the  azimuth  of  a 
terrestrial  mark  may  be  made  by  measuring,  with  a 
theodolite,  the  horizontal  angle  between  the  mark  and 
a  circumpolar  star  at  the  time  of  its  elongation,  i.e.,  its 
maximum  digression  from  the  meridian. 

It  may  be  seen  by  inspection  that  at  the  instant  of 
elongation  the  astronomical  triangle,  Fig.  3,  is  right- 
angled  at  the  star,  and  we  obtain  from  it, 

dsec  0, 


\  S  *-*) 
cos  tg  =  cot  d  tan  0, 

where  the  subscript  e  shows  that  the  azimuth  and  hour 
angle  are  those  at  elongation.  The  sidereal  time  of 
elongation  is  then  given  by 

(59) 


APPROXIMATE  DETERMINATIONS.  87 

where  the  upper  sign  is  to  be  used  for  the  western,  and 
the  lower  for  eastern  elongation.  ,If  D  denote  the  meas- 
ured angle  between  the  star  and  mark,  positive  when 
the  mark  is  east  of  the  star,  we  shall  have 

Azimuth  of  Mark  =  Ae+D.  (60) 

The  formulae  given  above  leave  nothing  to  be  desired 
in  respect  of  simplicity,  but  the  method  suffers  a  serious 
limitation  in  that  it  can  be  applied  only  at  certain  par- 
ticular times,  which  may  fall  at  very  inconvenient  hours 
of  the  day  or  night.  Polaris  is  the  star  most  frequently 
employed  in  this  way,  and  if  we  put  ie  equal  to  the  ex- 
pression derived  in  §  32  for  the  star's  hour  angle,  we 
shall  find  in  terms  of  the  A'  of  Equation  43,  for  the  local 
mean  solar  time  of  its  elongation, 

r-±*,-J'f  (61) 


the  upper  sign  for  the  western  elongation.  The  hour 
angle  tf  may  be  interpolated  from  the  following  short 
table  with  the  observer's  latitude  as  the  argument  : 

0        20°  30°  40°  50°  60° 

t.       5h  58m         5h  57m          5h  56m         5h  54m  5h  52m 

For  any  other  star  whose  polar  distance,  p,  is  less  than  5°  we  may 
assume  tf  =6h—  4  tan  $./>°.  This  formula  gives  the  last  term  in 
minutes  of  time  when  p°  is  expressed  in  degrees. 

Within  the  limits  of  the  United  States  if  Polaris  is 
observed  at  any  time  within  four  minutes  of  elongation 
its  true  azimuth  will  differ  from  its  azimuth  at  elonga- 
tion by  less  than  one  second  of  arc  ;  and  since  M  may  be 


88  FIELD  ASTRONOMY, 

found  by  the  above  formula  with  an  error  considerably 
less  than  four  minutes,  we  may  establish  the  rule : 

During  the  four  minutes  preceding  and  following  the 
time  M,  measure  the '  angle  {between  Polaris  and  the 
mark  an  equal  number  of  times  Circle  R.  and  Circle  L., 
being  careful  to  make  the  readings  of  the  azimuth  level 
the  same  in  the  two  positions  of  the  instrument. 
Reduce  the  mean  of  the  observations  as  if  it  were  a  single 
observation  made  at  elongation.  It  is  far  more  im- 
portant to  eliminate  instrumental  errors  by  suitable 
observations  in  both  positions  of  the  circle  than  to  make 
the  time  of  observation  agree  closely  with  the  com- 
puted time  of  elongation. 

Observations  of  stars  other  than  Polaris  may  be 
similarly  treated,  and  the  interval  from  elongation  within 
which  they  must  be  made  is  given  by  the  expression, 


T=  ±^-  .  cos  0sec  d,  (62) 

where  r  is  the  required  quantity,  in  minutes  of  time,  and 
xt  in  seconds  of  arc,  is  the  maximum  permissible  error 
in  the  result;  e.g.,  for  the  star  d  Ursas  Minoris,  observed 
at  elongation  in  latitude  43°,  we  adopt  as  the  limit  of 
negligible  error  x  =  2",  and  -find  r  =  ±  3-5m. 

To  plan  an  observation  of  d  Ursas  Minoris  to  be  made 
at  eastern  elongation  on  the  evening  of  May  14,  1902, 
at  a  place  whose  latitude  is  assumed  to  be  43°  4'  37", 
and  for  which  the  Standard  Time  in  common  use  is 
2.4m  slower  than  local  time,  we  proceed  as  follows: 


APPROXIMATE  DETERMINATIONS. 


89 


i 

sec  <j> 
cos  <5 
tan  0 

Ae 


43°  4' 37" 
86  36  45 

0.13642 
8.77150 


te 

86°  50' 

te 

5h47m-3 

a 

18      4    .0 

Be 

12    16    .7 

M 

8    49    .2 

Stand.  Time 

8    46    .8 

9 .97082 
184°  38'  24" 

From  the  standard  time  of  elongation  thus  deter- 
mined and  from  the  value  of  T  found  above,  it  appears 
that  observations  of  this  star  made  on  the  given  night 
at  any  time  between  8h  43 m  and  8h  50™,  standard 
time,  may  be  treated  as  if  made  at  elongation,  without 
introducing  into  the  result  an  error  greater  than  2". 

Corresponding  to  these  limits  we  have  the  following 
observations  of  the  angle  between  d  Ursae  Minoris  and 
the  mark  whose  azimuth  was  determined  in  §32.  The 
angle  was  measured  with  an  engineer's  transit,  by  the 
method  of  repetitions  (see  §  53),  including  only  two 
pointings  in  the  set,  with  a  reversal  of  the  instrument 
between  them.  After  the  reversal  the  bubble  of  the 
striding-level  was  brought  back  to  its  initial  position, 
as  shown  in  the  last  column  of  the  following  record,  by 
turning  the  levelling  screws,  and  the  mean  error  of  level 
is  thus  reduced  to  the  negligible  quantity, 


Wednesday,  May  14,  1902. 
At  Station  A.     Inst.  No.  306.     Observer,  C. 


Object. 

Circle. 

Point- 

Watch. 

Horizonta 

Circle. 

Levels. 

Vernier  A. 

Vernier,  B. 

SUrs.  Min. 
Mark  

L. 
R. 

I 
2 

h.      m. 
8      44 
8      40 

4  39   5 
251    24  o 

39      5 
24     o 

W.      E. 

9.8   12.2 
12.3   10  .0 

+  0.15 

90  FIELD  ASTRONOMY. 

We  find  from  the  above  record, 

24'  o"-4°  39'  5")  =  123°  22'  28". 


Azimuth  of  Mark  =  Aej-D      =308     o    52. 

Compare  this  result  with  the  rough  determination  of  the 
same  azimuth  made  in  §  32. 

38.  Time  and  Azimuth  from  Two  Stars.  —  An  excel- 
lent determination  of  time  may  be  made  by  measuring 
the  difference  of  azimuth  between  Polaris  and  a  southern 
star  and  noting  accurately  the  chronometer  times  of 
the  observations.  If  readings  to  a  terrestrial  mark  are 
combined  with  the  above  observations,  these  will  furnish, 
with  very  little  additional  labor,  a  good  determination 
of  the  azimuth  of  the  mark.  In  order  to  eliminate  the 
effect  of  instrumental  errors  from  the  resulting  time  and 
azimuth,  both  stars  and  mark  should  be  observed  in  each 
position  of  the  instrument,  i.e.,  Circle  R.  and  Circle  L., 
and  the  observations  should  be  arranged  symmetrically 
with  respect  to  time,  as  in  the  following  example. 

For  the  reduction  of  the  observations  we  recur  to 
Equation  14, 

cos  h  sin  A  =cos  d  sin  t,  (63) 

and  note  that  the  observed  (chronometer)  time  of  any 
observation,  T,  together  with  the  chronometer  correc- 
tion, AT,  and  the  right  ascension  of  the  star,  a,  suffice 
to  determine  its  hour  angle,  t,  through  the  relation, 

a  +  *  =  r  +  ^r.  (64) 

Similarly  the  azimuth  of  each  object  observed  will  differ 
from  the  corresponding  reading  of  the  horizontal  circle, 


PLATE  II. 


An  American  Theodolite'.     Diameter  of  Horizontal  Circle  8  inches. 
Approximate  Cost  $400. 

\Tofacep.  90.] 


APPROXIMATE  DETERMINATIONS.  91 

R,  by  a  constant  quantity,  AR,  which  is  called  the  index 
correction  of  the  circle,  so  that  we  shall  have, 

A=R-AR.  (65) 

The  altitude,  h,  may  be  determined  directly  from 
readings  of  the  vertical  circle,  and  if  all  of  the  above 
quantities  are  correctly  known,  their  values  when  intro- 
duced into  Equation  63  will  satisfy  it.  If  they  do  not 
satisfy  the  equation,  something  must  be  wrong  with  the 
assumed  values  and  we  proceed  to  find  from  Equation  63 
a  means  of  correcting  the  assumed  AT  and  AR  so  that 
they  shall  satisfy  the  equation.  For  this  purpose  let 
u  and  m  represent  values  of  AT  and  AR  provisionally 
assumed,  and  made  as  nearly  correct  as  can  be  con- 
veniently estimated,  e.g.,  within  2m  and  30'  respectively, 
and  denote  by  %  and  y  the  unknown  corrections  which 
must  be  added,  algebraically,  to  u  and  m  in  order  to 
obtain  the  true  chronometer  correction  and  the  true 
index  correction  of  the  azimuth  circle.  We  shall  then 
have  for  each  star, 


(66) 
and  introducing  into  these  equations  the  abbreviations, 


we  find  that,  in  terms  of  these  symbols,  Equation  63 
takes  the  form 

cos  h  sin  (p  —  y)  +cos  d  sin  (r—x)  =o.  (67) 
When  x  and  y  are  quantities  as  small  as  is  above  sup- 
posed, their  squares  and  higher  powers  may  be  neglected 


92  FIELD  ASTRONOMY. 

without  producing  errors  greater  than  10",  and  these 
errors,  being  multiplied  by  the  factor  sin  A  or  its  equiva- 
lent, will  be  reduced  to  less  than  i"  whenever  the  stars 
observed  are  within  6°  of  the  meridian,  in  azimuth. 
Assuming  that  the  observations  will  be  limited  to  stars 
near  the  meridian,  we  may  put 

sin  x=x  .  sin  i",    siny=y  .  sin  i",    cos  x  =  cosy  =  i, 

and  expanding  Equation  67  and  introducing  these  values 
we  find, 

cos  h  sin  //  —  cos  h  cos  /*  .  y  .  sin  i" 

+  cos  d  sin  T  —  cos  d  cos  T  .  x  .  sin  i"  =o.        (68) 

Dividing  this  equation  through  by  the  factor, 

15  sin  i"  cos  h  cos  //, 
and  introducing  the  abbreviation, 

g  =  cos  d  cos  T  sec  h  sec  /z, 
we  obtain  in  place  of  Equation  68  the  relation, 


y  +  g%  =  1  5  sin  i"  (tan  ft  +  g  tan  T)* 
Each  star  observed  will  furnish  an  equation  of  this  form, 
in  which  the  second  member  contains  only  known  quan- 
tities, and  from  observations  of  two  stars  we  may  there- 
fore determine  the  two  unknown  quantities  x  and  y. 
These  will  be  furnished  by  the  solution,  expressed  in 
seconds  of  time  as  the  unit,  on  account  of  the  factor 

—  -.  -  77.     We  shall  hereafter  represent  this  factor  by 
15  sin  i 

the  letter  G    and  employ  the  numerical  value, 
log  G  =  4.13833. 


APPROXIMATE  DETERMINATIONS.  93 

It  is  sometimes  convenient  to  use,  instead  of  the  equation  above 
given  for  g,  another  form  in  which  the  altitude,  h,  does  not  appear 
explicitly,  since  it  will  then  be  feasible  to  omit  the  observation  of  the 
stars'  altitudes  and  thus  simplify  the  observing  programme.  For 
this  purpose,  assuming  that  t=  —  T,  we  readily  find  from  Equations 
14  the  relation, 


coslcos      =  sin  0  -      c?s         ,  (70) 

cos  d  cos  T  cot  d  cos  T 

in  which  the  left-hand  member  is  the  reciprocal  of  g.     If  we  now  in- 
troduce the  auxiliary  quantity, 

cot  D  =cot  d  cos  r,  (71) 

we  shall  find  in  terms  of  the  new  auxiliary  , 

cos  D 


39.  Effect  of  Erroneous  Levelling.  —  In  the  preceding 
analysis  it  is  tacitly  assumed  that  the  instrument  was 
perfectly  levelled,  and  the  observer  should  seek  to  fulfil 
this  condition  as  nearly  as  possible.  After  finishing 
the  observation  Circle  R.  and  while  the  telescope  is  still 
pointing  south,  read  the  striding  level  and  record  the 
position  of  the  bubble  in  its  tube.  Then,  without  revers- 
ing the  level  on  the  axis,  reverse  the  instrument,  to 
Circle  L.,  and  by  means  of  its"  levelling  screws  bring  the 
bubble  back  to  its  former  place  in  the  tube,  i.e.,  to  the 
same  scale  reading.  By  this  process  the  vertical  axis 
will  be  as  much  out  of  plumb  in  the  one  direction,  east, 
for  Circle  L.,  as  it  was  out  in  the  other  direction,  west, 
for  Circle  R.,  and  these  errors  will  compensate  each  other 
in  the  mean  result. 

It  will  sometimes  happen,  however,  that  the  readings  of  the  bubble 
Circle  R.  and  Circle  L.  will  be  appreciably  different  and  then  the  average 
inclination  of  the  vertical  axis  to  the  plane  of  the  meridian,  which 
we  shall  represent  by  b',  must  be  determined  from  the  level  readings, 
as  shown  in  §  42.  To  determine  the  effect  of  this  error  upon  the  com- 
puted #  and  y  we  have  recourse  to  Fig.  6,  which  represents  a  part  of 


'o*SJ 


94 


FIELD  ASTRONOMY. 


the'celestial  sphere,  where  P  and  Z  are  respectively  the  pole  and  zenith, 
T  is  the  terrestrial  mark  whose  azimuth  is  to  be  determined,  and  B 
is  the  point  of  the  sphere  toward  which  the  mean  position  of  the 
vertical  axis  was  directed.  The  observations,  uncorrected  for  level 
error,  have  determined  through  x  and  y  the  chronometer  correction 
and  the  azimuth  of  T  referred  to  the  meridian  BPH'  instead  of  to 


FIG.  6.— Effect  of  Level  Error. 

the  true  meridian  ZPH,  and  the  error  in  AT  is  therefore  the  angle,  /?, 
between  the  two  meridians.  Similarly  the  error  in  the  computed 
azimuth  of  T  is  the  arc  of  the  horizon,  HH',  intercepted  between  the 
two  meridians.  From  the  figure  we  find, 


4x=p  =  b'  sec  <}>, 

Ay  =H  -Hf  =/?  sin  <j>  =b'  tan 

which  are  the  required  level  corrections. 


(73) 


40.  Example. — We  now  collect  and  slightly  rearrange 
our  formulae  in  the  following  group  of  equations  which 


APPROXIMATE  DETERMINATIONS.  95 

are  to  be  used  in  the  actual  reduction  of  a  set  of  observa- 
tions : 

Compute  for  each  star  the  following  quantities  : 


log  £  =  4.13833,        cot  D  =  cot  d  COST,          , 

cos  d  cos  T  cos  D 

&  =  cosh  cos  n  £  =  sin(0-£>)' 

=  G  tan  jL  +    G  tan  r. 


From  the  equations  of  this  type  furnished  by  two  or 
more  stars  find,  by  an  algebraic  solution,  the  values  of 
x  and  y,  and  from  these  values 

x  —  b'  sec  0), 

15(7  +  6'  tan  0).  (75' 

The  level  constant,  6',  positive  when  the  vertical 
axis  is  tipped  toward  the  east,  is  here  supposed  to  be 
expressed  in  seconds  of  time,  and  the  coefficient  15  in 
the  last  equation  transforms  both  y  and  br  from  time 
into  arc.  The  azimuth  of  the  mark  T  is  to  be  obtained 
by  applying  the  index  correction,  J.R,  to  the  mean  of 
the  circle  readings  to  the  mark. 

For  stars  within  15°  or  20°  of  the  equator  we  may 
usually  substitute  in  place  of  the  formulae  given  above 
the  simpler,  approximate  expressions, 

D  =  dt         G  tan  /z  =  //  (in  seconds  of  time)  .          (76) 

The  following  example  shows  the  record  and  reduc- 
tion of  such  a  determination  of  time  and  azimuth,  made 
with  an  engineer's  transit: 


96 


FIELD  ASTRONOMY. 


AZIMUTH   OF    STATION  B. 

Monday,  April  14,  1902. 

At  Station  M.     Engineer's  Transit,  F.     Chronometer,  S. 
Observer,  ^C. 


Object. 

Circle. 

Chronometer 
Time. 

Horizontal  Circle. 

Vertical 
Circle. 

Levels. 

Vernier  A. 

Vernier  B. 

Sta.  B.  . 
Polaris.  . 
e  Corvi   . 
e  Corvi.  . 
Polaris.  . 
Sta.  B.. 

L. 
L. 
L. 
R. 
R. 
R. 

h.     m.        s. 
II    55    — 

ii   57   10 

12        I        7.6 

12     8     7.9 

12     13     17 
12     l6    

0           /           // 

7  55  45 
359  20     o 
178   56  50 
o  43  40 
179   26   25 
187   55  35 

55     30 
19     5° 
56     i5 
43     3° 
25     45 
55     20 

0               / 

42     o 

24  55 
24  52 

4i  55 
o     o 

W.         E. 
6.1     4.2 
4-9     6.6 

fe'  =  +0.4  d 
d=Z" 

Star. 
6 
a 
T 

T+u 

r  (Ire) 
cot  d 

COS  T 

D 

cos  D 
sin(<£-£>) 

g 

tan  T 

G 

tan  fj. 

g  G  tan  T 

G  tan  fj. 


REDUCTION. 

Polaris. 

e  Corvi. 

88°  47'    8" 

-22°    4'  49" 

Ih  22m  58' 

i2h   s-    7-.8 

12       5      14 

12     4     37-7 

12       5       34 

12     4     57-7 

13     17      24 

O      O       IO.I 

I99°  21'      0" 

0°    2'       32" 

179    23      o 

359    5°     4 

8.3263 

0.3919?* 

9.9748?* 

o.oooo 

91°  8'  45" 

-2  2°  4'  49" 

8.  3009?* 

9.9670 

9.8716?* 

9-9579 

8.4293 

o  .  0091 

9-5455 

6.8674 

4-1383 

4-1383 

8.0319?* 

7.4608?* 

+  129.7 

+    iQ-35 

—  148.0 

-  39-73 

u 

x 

AT 


m 


Sta.  B 
Az. 


43°  4'  37" 


+  0m  20. 80 
—  O       II.  2 

+  o       8.  8 


ou      o'     o' 

-4    30 

—  o        4'  30' 

187  55    32 

188  o      2 


Equations. 
^  +  0.027*=  —18.3 
y  +  i.o2ix=-29.4 


We  note  in  connection  with  the  record  of  these 
observations  that  immediately  after  the  reversal  from 
Circle  L.  to  Circle  R.  the  instrument  was  relevelled  as 
suggested  in  §  39,  and  the  resulting  mean  inclination 
of  the  vertical  axis,  6' =  +  1.2",  is  so  small  that  it  is 
neglected  in  the  reduction.  Had  the  level  corrections 
been  retained,  they  would  have  altered  the  values  of 


APPROXIMATE  DETERMINATIONS.  97 

AT  and  the  azimuth  of  Station  B  by  —  os.i  and  +i" 
respectively.  Approximate  readings  of  the  vertical 
circle  (one  vernier,  to  the  nearest  minute)  are  contained 
in  the  record,  and  from  them  the  value  of  g  for  each  star 
may  be  computed  if  the  latitude  is  supposed  unknown. 

The  assumed  values  of  u  and  m  are  obtained  as  fol- 
lows :  Since  e  Corvi  was  observed  near  the  meridian  and 
the  mean  of  the  times  recorded  for  it  does  not  differ  much 
from  the  star's  right  ascension,  it  is  evident  that  the 
chronometer  correction  is  small,  and  neglecting  this  cor- 
rection, i.e.,  treating  the  observed  times  as  true  sidereal 
times,  we  obtain  the  hour  angle  of  Polaris  and,  by  Table  I, 
its  azimuth  at  the  time  of  observation.  This  azimuth 
proves  to  be  about  the  same  as  the  corresponding  circle 
readings,  the  instrument  had  been  roughly  oriented, 
and  we  therefore  assume  m  =  o.  Returning  to  the  obser- 
vations of  s  Corvi,  and  when  necessary  subtracting  m 
from  the  circle  readings,  we  find  by  interpolation  be- 
tween the  two  observations  that  the  chronometer  time 
corresponding  to  the  corrected  circle  reading  o°  o',  i.e.,  to 
the  star's  meridian  transit,  was  approximately  i2h  5m, 
agreeing  so  closely  with  the  right  ascension  of  the  star 
that,  to  the  nearest  minute,  we  might  assume  u  =  o.  For 
the  sake  of  illustration,  however,  a  slightly  different 
value  is  adopted  in  the  reduction.  This  method  will 
always  furnish  sufficiently  accurate  results  for  u  and  m 
if  the  instrument  is  approximately  oriented  before  be- 
ginning the  observations.  When  the  time,  T,  is  taken 
from  a  watch,  i.e.,  solar  time,  u  may  be  a  large  number, 
e.g.,  anything  from  o  to  i2h. 


98  FIELD  ASTRONOMY. 

41.  Subsidiary  Determination  of  Time. — The  method 
of  §  40  is  especially  convenient  when  the  accurate  azi- 
muth of  a  mark  is  to  be  determined  and  the  time  is 
required  only  for  this  purpose".  The  angle  between  the 
polar  star  and  the  mark  should  then  be  measured  as 
shown  in  §  53,  while  one  or  more  measurements,  Circle  R. 
and  Circle  L.,  of  the  angle  between  the  mark  and  a  south- 
ern star  near  the  meridian,  will  determine  AT  with  all 
needful  precision  and  with  a  minimum  expenditure  of 
labor. 

The  reduction  of  these  observations  will  differ  from 
the  method  given  above  only  in  the  following  respects: 
We  here  put  m  equal  to  an  assumed  approximate  azi- 
muth of  the  mark,  represent  by  L  the  measured  angle 
between  the  mark  and  star,  and  compute,  for  each  star, 
the  quantity  //  from  the  formula 

H  =  m±LJ  (77) 

using  the  upper  sign  when  the  azimuth  of  the  star  is 
greater  than  that  of  the  mark.  The  quantities  x,  y,  and 
AT  are  then  to  be  determined  as  above  and  the  resulting 
azimuth  of  the  mark  will  be 

AM=m-i$y.  (78) 


CHAPTER  VII. 

INSTRUMENTS. 

IN  the  several  determinations  thus  far  considered 
we  have  for  the  most  part  assumed  that  the  data  fur- 
nished by  the  instruments  employed  were  free  from 
purely  instrumental  errors,  and  in  approximate  work 
this  may  usually  be  done  if  due  care  has  been  bestowed 
upon  the  adjustments.  But  where  a  higher  degree  of 
precision  is  required  it  becomes  necessary  to  study  the 
instrument  employed,  as  being  in  itself  a  source  of  errors 
that  need  to  be  eliminated,  and  we  must  turn  therefore 
to  a  more  detailed  consideration  of  some  of  the  instru- 
ments used  in  field  astronomy  before  taking  up  the  class 
of  methods  called  accurate. 

42.  The  Spirit-level.  —  The  spirit-level  is  used  in 
astronomical  practice  to  measure  small  deviations  of  a 
line  or  surface  from  a  vertical  or  horizontal  position, 
and  incidentally  to  adjust  a  part  of  an  instrument  to 
such  a  position.  It  consists  essentially  of  a  glass  tube 
bent  or  ground  into  an  arc  of  a  circle  of  large  radius  and 
so  mounted  that  the  plane  of  this  circle  is  approximately 
vertical.  The  tube  being  nearly  filled  with  ether  and 
its  ends  hermetically  sealed,  the  small  volume  of  air  or 


100  FIELD  ASTRONOMY. 

vapor  that  remains  in  the  tube  is  collected  into  a  bubble 
which  always  stands  at  the  highest  point  of  the  circle, 
so  that  a  line  drawn  from  its  middle  point  through  the  cen- 
tre of  curvature  of  the  tube  is  vertical.  The  upper 
surface  of  the  tube  is  usually  provided  with  a  scale  of 
equal  parts,  and  the  position  of  the  bubble  in  the  tube 
is  determined  by  the  readings  of  its  ends  upon  this  scale. 
The  angle  subtended  at  the  centre  of  curvature  of  the 
tube  by  the  space  between  two  consecutive  lines  is  called 
the  value  of  a  division  of  the  level,  and  this  value,  which 
will  be  represented  by  2d,  is  required  for  transforming 
the  indications  of  the  level  into  seconds  of  arc.  Note 
that  d  represents  one  half  the  value  of  a  level  division. 

Let  such  a' level  be  supposed  attached  to  a  theodolite, 
the  inclination  of  whose  vertical  axis  to  the  true  vertical 
is  to  be  determined  from  readings  of  the  bubble.  We 
are  here  concerned  with  angular  measurements,  e.g.,  the 
angle  that  the  axis  makes  with  the  true  vertical;  the 
angle  moved  over  by  the  level  bubble,  as  seen  from  the 
centre  of  curvature  of  the  tube,  when  the  instrument 
is  turned  from  one  position  to  another;  etc.,  and  as  the 
simplest  method  of  dealing  with  these  angles  we  shall 
imagine  the  whole  apparatus  projected  radially  upon 
the  celestial  sphere,  so  that  the  arc  joining  the  points 
in  which  any  two  projected  lines  meet  the  sphere,  meas- 
ures the  angle  between  these  lines.  This  method  of 
analysis  by  projecting  the  parts  of  an  instrument  upon 
the  sphere  is  in  common  use,  and  the  student  should 
acquire  a  clear  conception  of  the  simple  case  to  which 
it  is  here  first  applied. 


INSTRUMENTS.  101 

To  determine  the  relation  of  the  bubble  readings  to 
the  required  inclination,  we  imagine  the  axis  of  the  in- 
strument and  the  plane  of  the  level  extended  until  they 
meet  the  celestial  sphere,  as  in  Fig.  7,  which  represents 


L 

FlG.  7.— Theory  of  the  Spirit-level. 

a  small  part  of  the  sphere  adjacent  to  the  zenith,  Z. 
In  this  figure  V  is  the  point  in  which  the  produced  axis 
meets  the  sphere,  and  LB  is  the  trace  of  the  plane  of  the 
level  tube  upon  the  sphere.  The  projection  of  the  mid- 
dle of  the  bubble  upon  the  sphere  must  be  at  B}  the  point 
in  LB  nearest  to  Z,  and  found,  therefore,  by  letting  fall 
a  perpendicular  from  Z  upon  LB.  If,  now,  the  theod- 
olite be  turned  180°  in  azimuth,  i.e.,  rotated  about  V 
as  a  pivot,  the  level  tube  will  be  revolved  about  V  as  a 
centre,  into  the  position  L'B',  and  the  point  B  will  fall 
at  Bf,  but  the  middle  of  the  bubble  will  stand  at  B" 
instead  of  Bf ',  since  this  is  now  the  point  nearest  to  the 
zenith.  From  elementary  geometrical  considerations, 
VM  =  \B'B",  where  B'B"  is  the  space  moved  over  by 
the  level  bubble  when  the  instrument  is  turned  from 


102  FIELD  ASTRONOMY. 

one  position  to  the  other,  and  VM  is  the  projection 
upon  the  plane  of  the  level  tube  of  the  arc,  VZ,  that 
measures  the  angle  between  the  axis  of  rotation  and 
the  true  vertical.  Calling  this  projection  b  and  repre- 
senting by  a',  6',  a",  b"  the  scale  readings  of  the  ends 
of  the  bubble  in  the  two  positions,  we  have 


'\     ,      (q'-6")  +  (fr'-q'% 
)2d=  —         ~d'     (79) 


It  is  customary  to  record  the  several  readings  in  the 
form, 

(Symbols  used  above.)       (Actual  observations.) 

N          S  N  S 

a'        b'  16.4        32.2 

a"    ;  b"  39.6         9.1 

7-35 

The  letters  N  and  S  denote  the  north  and  south  ends 
of  the  level  tube,  or  some  equivalent  system  of  distinguish- 
ing between  them. 

43.  Discussion  of  the  Level   Readings.  —  The    student 
should  now  note  that  : 

(a)  The  coefficient  of  d  in  Equation  79  is  the  mean  of 
the  diagonal  differences  in  the  square  array  formed  by 
the  four  numbers  tabulated  in  the  preceding  example. 
This  example  represents  the  manner  in  which  level  read- 
ings should  be  recorded,  and  the  mean  of  the  diagonal 
differences,  7.35,  written  below  the  line,  should  be  worked 
out  and  entered  with  the  record. 

(b)  If  the  bubble  readings  have  been  correctly  taken 
and  there  is  no  change  in  the  length  of  the  bubble  during 
the  observation,  these  differences  must  be  equal,  one 


INSTRUMENTS.  103 

( 
to  the  other,  thus  furnishing  a  check  upon  the  accuracy 

of  the  level  readings,  which  should  always  be  applied 
immediately  after  recording  them.  If  the  temperature 
is  changing  rapidly,  the  length  of  the  bubble  may  be 
changed  and  the  check  impaired  without  necessarily 
diminishing  the  accuracy  with  which  b  is  determined. 

(c)  If  the  greatest  of  the  four  numbers  stands  in  the 
column  marked  N,  the  north  end  of  the  level  tube  is  on 
the  whole  higher  than  the  south  end,  and  the  vertical 
axis  is  tipped  toward  the  south.     Determine  the  sign 
of  b  in  this  manner. 

(d)  The  zero  of  a  level  graduation  is  sometimes  placed 
at  one  end  of  the  scale  and  sometimes  in  the  middle,  but 
the  method  of  record  and  reduction  given  above  applies 
to  both  cases. 

(e)  It  is  apparent  from  the  figure  that  the  point  of 
the  level  tube  midway  between  B'  and  B"  marks  that 
radius  of  the  level  tube  which  is  most  nearly  parallel 
with  the  rotation  axis  of  the  instrument.      Since  this 
radius  ought  to  pass  through  the  middle  point  of  the 
scale,  and  does  so  pass  when  the  level  is  in  adjustment, 
we  have  as  the  error  of  adjustment  of  a  level  numbered 
continuously  from  one  end  to  the  other, 


(80) 


where  5  represents  the  total  number  of  divisions  in  the 
level  scale.  In  the  example  given  above  5  =  50  and 
€  =  0.7  division. 

The  essential  element  in  the  determination  of  b  is 


104  FIELD  ASTRONOMY. 

the  reversal  of  the  level  with  the  resulting  displace- 
ment of  the  bubble,  and  it  is  a  matter  of  indifference 
whether  this  displacement  is  produced  by  revolving 
the  level  about  a  vertical  axfs  to  which  it  is  attached, 
as  in  the  case  considered  above,  or  by  picking  the  level 
up  bodily  from  a  plane  or  line  upon  which  it  stands, 
turning  it  end  for  end  and  replacing  it  in  the  reversed 
position,  as  is  done  in  measuring  the  inclination  of  an 
approximately  horizontal  axis.  Let  the  student  show 
that  the  inclination  of  this  axis  to  the  plane  of  the 
horizon  may  be  obtained  from  the  bubble  readings 
exactly  as  the  inclination  of  the  vertical  axis  was  deter- 
mined above.  The  greatest  of  the  four  readings  is 
adjacent  to  the  high  end  of  the  axis.  Determine  in  this 
way  the  inclination  of  the  horizontal  axis  of  a  theodolite. 
A  fine  level  is  an  exceedingly  sensitive  instrument 
and  requires  great  care  in  its  use.  Unless  unusually  well 
supported  its  readings  may  be  vitiated  by  the  observer 
passing  from  one  side  of  it  to  the  other,  or  even  by  shift- 
ing his  weight  from  one  foot  to  the  other.  Therefore 
observe  the  following  precepts: 

1 .  Keep  away  from  the  level  as  much  as  possible. 

2.  Don't  allow  the  sun  to  shine  upon  it. 

3.  Don't  hold  a  source  of  heat,  e.g.,  a  lamp  or  your 
own  hand,  near  a  level  longer  than  is  strictly  necessary. 

4.  If  the  level  has  a  chamber  with  reserve  supply  of 
air  at  one  end  of  the  tube,  use  it  to  regulate  the  length 
of  the  bubble,  keeping  this  always  about  one  half  as 
long  as  the  scale. 

5.  Make  the  inclinations  that  are  to  be  measured 


INSTRUMENTS.  105 

as  small  as  possible,  in  order  to  avoid  any  considerable 
run  of  the  bubble  and  the  resulting  effect  of  possible 
irregularities  in  the  level  tube. 

44.  Value  of  a  Level  Division. — The  value  of  a  level 
division  is  most  conveniently  determined  by  measuring 
with  a  micrometer,  or  finely  graduated  circle,  the  vertical 
angle  through  which  its  tube  must  be  tipped  in  order 
to  cause  the  bubble  to  run  past  a  given  number  of  divi- 
sions of  the  scale.  If  the  necessary  apparatus  for  such 
a  determination  is  not  at  hand,  the  following  method 
will  furnish  equally  good  results  and  requires  only  an 
engineer's  transit,  to  which  the  level  must  be  attached 
with  its  plane  approximately  vertical.  ,-. 

Let  the  instrument  be  firmly  set  up  but  very  much 
out  of  level,  e.g.,  with  its  vertical  axis  making  an  angle, 
7-,  with  the  true  vertical  amounting  to  2°,  more  or  less. 
See  p.  109  for  a  method  of  determining  the  exact  value 
of  this  angle,  which  will  be  required  in  the  reduction  of 
the  observations.  If  the  transit  is  now  turned  slowly 
about  its  vertical  axis  (azimuth  motion),  the  level -bubble 
will  run  back  and  forth  in  its  tube,  and  two  positions  of 
the  instrument  may  be  found  at  which  the  bubble  will 
come  to  the  middle  of  its  scale.  We  shall  designate  the 
readings  of  the  azimuth  circle  corresponding  to  these  two 
positions  by  Al  and  A2. 

Any  slight  turning  of  the  instrument  either  way 
from  A!  or  A  2  will  cause  a  corresponding  slight  motion 
of  the  bubble,  and  to  determine  the  relation  of  the  bubble 
readings  to  the  corresponding  circle  readings  we  resort 
to  Fig.  8,  which  represents  a 'portion  of  the  celestial 


106  FIELD  ASTRONOMY. 

sphere  adjacent  to  the  zenith,  Z.  V  is  the  point  in 
which  the  deflected  axis  of  the  instrument  meets  the 
sphere,  and  SV  is  the  trace  upon  the  sphere  of  the  plane 
of  the  level-tube,  which  is  assuirted  to  have  been  adjusted 
approximately  parallel  to  the  vertical  axis  of  the  transit. 
Small  errors  in  this  adjustment  are  of  no  consequence. 


FIG.  8.  —  Determination  of  d. 

As  the  instrument  is  turned  in  azimuth,  carrying  the 
level-tube  with  it,  the  arc  SV  must  revolve  about  V  as 
a  pivot,  and  the  amount  of  its  rotation  will  be  measured 
by  the  successive  readings  of  the  azimuth  circle.  It 
may  be  seen  readily  that  the  angle  T  of  the  figure  corre- 
sponding to  any  particular  circle  reading,  A,  is  given 
by  the  equation 

-^,  (82) 


being  the  circle  reading  at  which  SV  coincides 
with  VZ. 


INSTRUMENTS.  107 

Since  a  level-bubble  always  stands  at  the  highest 
point  of  its  tube,  the  point  nearest  the  zenith,  we  may 
find  the  point  in  the  figure  corresponding  to  the  middle 
of  the  level-bubble  by  drawing  from  Z  an  arc  of  a  great 
circle  perpendicular  to  SF,  and  the  intersection,  5,  will 
be  the  required  point.  In  the  right-angled  spherical 
triangle  SVZ  thus  formed  we  have  the  relation, 

tan  p  =tan  7-  cos  T,  (83) 

in  which  p  measures  the  distance  of  the  middle  of  the 
bubble  from  the  fixed  point  V.  To  find  the  effect  upon 
p  of  any  small  variation  in  T,  i.e.,  to  find  how  far  the 
bubble  will  run  when  the  instrument  is  turned  slightly 
in  azimuth,  we  differentiate  this  equation  and  obtain 

—  dp  =  tan  f  cos2  p  sin  T  dr,  (84) 

and  substituting  in  place  of  these  differentials,  small 
finite  increments  of  the  respective  quantities,  we  obtain 

2d(b'-b")  =tan  r  cos2  p  sin  T  (A' -A"),         (85) 

where  d  represents  the  value  of  half  a  level  division  and 
bf  and  b"  are  the  scale  readings  of  the  middle  of  the  bubble, 
corresponding  to  the  circle  readings  A'  and  A" . 

Equation  85  may  be  used  to  determine  the  value  of  d, 
but  whenever  ordinary  care  is  bestowed  upon  the  ad- 
justment of  the  level,  i.e.,  to  make  the  radius  passing 
through  the  middle  point  of  the  scale  parallel  to  the  ver- 
tical axis  of  the  theodolite  (Equation  80),  the  readings 
Al  and  A2  will  be  so  nearly  180°  apart  that  we  may  put 
r  =  9o°,  cos/?  =  i,  for  all  positions  of  the  bubble  within 


108  FIELD  ASTRONOMY. 

the  limits  of  its  scale,  and  thus  obtain  in  place  of  Equa- 
tion 85  the  simpler  relation 

A'—  A" 

-  (86) 


In  this  equation  A'  -A"  and  bf  —  b"  are  to  be  derived 
from  the  readings  of  the  horizontal  circle  and  level, 
respectively,  and  in  making  observations  for  their  deter- 
mination it  is  well  to  bring  the  bubble  as  near  as  may 
be  to  one  end  of  the  tube  and  set  the  circle  to  read  the 
nearest  integral  10'.  Then  turn  the  instrument  to  each 
successive  10'  or  20'  reading,  and  record  the  readings 
of  the  bubble  until  the  former  has  traversed  the  entire 
length  of  its  scale,  after  which  repeat  the  operation  in 
the  inverse  order,  using  the  same  circle  settings  as  before. 
With  reference  to  the  direction  of  the  bubble's  motion 
these  two  series  will  be  designated  as  Forward  and 
Backward.  Having  completed  these  observations,  turn 
the  instrument  to  the  second  position  in  which  the  bubble 
plays,  e.g.,  from  Al  to  A2,  and  make  a  similar  double  set 
of  readings. 

The  readings  obtained  at  any  two  settings  of  the 
instrument  will  determine  values  of  A'  —  A"  and  bf  —  b"  , 
and  therefore  a  value  of  d,  but  it  is  advisable  to  secure 
a  considerable  number  of  these  determinations,  ranging 
over  the  whole  length  of  the  level-tube,  in  order  to  test 
its  uniformity.  Supposing  such  a  series  to  have  been 
made,  the  manner  of  forming  the  differences  b'  —  b" 
illustrated  below,  may  be  followed  with  advantage,  i.e., 
subtract  the  first  b  from  the  first  one  following  the 


INSTRUMENTS.  109 

middle  of  the  set,  the  second  b  from  the  second  one  after 
the  middle,  etc. 

The  angle  7-  of  Equation  86  should  be  determined 
at  the  time  of  deflecting  the  axis,  as  follows :  After  having 
carefully  levelled  the  instrument,  take  a  reading  of  the 
vertical  circle  when  the  line  of  sight  is  directed  toward 
a  fixed  mark,  that  we  may  call  P.  By  means  of  the 
levelling  screws  deflect  the  axis  exactly  toward  or  from 
P  through  some  convenient  angle,  e.g.,  i°  if  the  vertical 
circle  reads  to  seconds,  3°  if  it  reads  only  to  minutes,  and 
again  point  upon  P  and  read  the  circle.  The  difference 
of  the  two  readings  is  the  value  of  7-.  To  make  sure  that 
the  deflection  of  the  axis  is  made  in  the  proper  direction, 
by  means  of  the  levelling  screws  make  the  reading  of 
the  azimuth  level  (see  §  50)  the  same  after  deflection 
that  it  was  before  deflection,  and  there  will  then  be  no 
component  of  deflection  perpendicular  to  the  direction  P. 

45.  Example.  —  We  have  the  following  example  of 
the  record  and  reduction  of  the  first  half  of  a  complete 
set  of  observations  for  the  determination  of  d.  In  the 
reduction  we  note  that  the  divisor  2(6'  —  b")  of  Equation 
86  is  equivalent  to  26'  —  26",  and  since  b  is  the  scale  read- 
ing of  the  middle  of  the  bubble,  26  is  equal  to  the  sum 
of  the  readings  of  the  ends  of  the  bubble.  The  column 
headed  26  is  found  in  this  way  from  the  mean  of  the  two 
sets  of  bubble  readings  opposite  each  circle  reading. 

The  regular  progression  of  the  numbers  in  the  column 
2(6'  —  b")  suggests  a  level-tube  of  variable  curvature, 
but  the  amount  of  data  is  not  sufficient  to  decide  this 
with  certainty.  More  observations  are  needed. 


110  FIELD  ASTRONOMY. 

Friday,  Dec.  7,  1894. 

Alidade  Level  of  Universal  Instrument,  No.  2598. 
Readings  to  Mark. 
.Mic.  I.  Mic.     II.        -  Mean. 

Axis  Vertical 180°'  26'  45"  •*         26'    45  180°  26'  45" 

Axis  Deflected 179270  27     6  179    27     3 


y  =       o    59  42 
Bubble. 
Azimuth  Circle.  Forward.  Backward. 

«/;    „          ~    „  26 . 0         0.4 

27.7          1.9 
29.7          3.9 

32.1       6.2 

34-2        8.5 
36.5     10.6 

30'  =1800" 
8.  239 
3-  255 


A'  -A" 

log  tan  f 

log(A'  -A" 

colog  2  (&'-&") 

log  d 
46.75        12.75  d 


o.  401 

2.    52 


12  .40 

46.  Inequality  of  Pivots. — When  a  spirit-level  is  used 
to  determine  the  inclination  of  a  line,  such  as  the  hori- 
zontal axis  of  a  transit,  its  readings  and  the  resulting 
inclination  will  be  vitiated  by  any  inequality  which  may 
exist  in  the  diameter  of  the  pivots  upon  which  it  rests. 
To  test  for  such  an  inequality  let  the  instrument  be 
firmly  mounted  and  the  inclination,  6',  be  measured 
with  the  level  as  shown  in  §  42 ;  then  lift  the  axis  out 
of  the  wyes,  turn  it  end  for  end,  and  replace  it  so  that 
what  was  the  east  pivot  shall  now  rest  in  the  west  wye. 
Again  measure  the  inclination,  b",  and  repeat  the  level- 
ings  and  reversals  several  times,  so  that  any  systematic 
difference  which  may  exist  between  b'  and  b"  shall  be 
well  determined.  We  now  put 

b"\  (87) 


INSTRUMENTS.  Ill 

where  i  is  the  correction  for  inequality  of  pivots,  and 
find  for  the  true  inclination  of  the  axis  in  the  two  posi- 
tions, 

b^b'-i,  b2  =  b"  +  i.  (88) 

The  correction  i  should  be  carefully  determined  and 
applied  to  all  measured  values  of  the  inclination. 

47.  The  Theodolite.  —  This  instrument,  which  is  also 
called  engineer's  transit,  altazimuth,  universal  instru- 
ment, etc.,  is  one  with  whose  general  appearance  and 
construction  the  student  is  supposed  sufficiently  familiar 
to  recognize  its  close  relationship  with  the  coordinates 
of  System  I,  altitude  and  azimuth.  Trace  out  this  rela- 
tionship in  Plates  I,  II,  and  III,  which  represent  different 
types  of  this  instrument.  The  line  of  sight  (telescope) 
is  a  radius  vector  of  undetermined  length;  the  hori- 
zontal and  vertical  circles  measure  azimuths  and  alti- 
tudes, or  zenith  distances,  and  in  an  ideally  perfect 
instrument  the  readings  of  these  circles  should  be  the 
true  azimuth  and  altitude  of  the  line  of  sight,  or  at 
most  should  differ  from  these  only  by  a  constant  index 
correction. 

It  may  readily  be  seen  that  among  the  conditions 
which  must  be  satisfied  in  the  construction  of  such  an 
instrument  are  the  following: 

(1)  The  axes  must  be  perpendicular  to  each  other. 

(2)  The  line  of  sight  must  be  perpendicular  to  the 
horizontal  axis. 

(3)  The  vertical  axis  must  be  truly  vertical. 
Owing  to  unavoidable  imperfections  of  mechanical 


112  FIELD  ASTRONOMY. 

work  it  is  not  probable  that  any  one  of  these  conditions 
is  exactly  fulfilled  in  any  given  instrument,  and  they 
are  therefore  to  be  regarded  as  so  many  sources  of  error, 
whose  effects  may  be  made  small  by  careful  adjustment, 
but  whose  complete  elimination  must  be  sought  in  some 
other  way;  e.g.,  if  the  vertical  axis  is  not  truly  vertical, 
we  may  determine  as  follows  the  means  for  correcting 
the  effect  of  this  error  upon  the  measurement  of  altitudes. 
48.  Zenith  Distances. — In  Fig.  9  let  HZ  be  the  direc- 


FIG.  9. — Measurement  of  Zenith  Distances. 

tion  of  the  vertical;  5,  a  point  whose  zenith  distance, 
ZHS,  is  to  be  determined;  HV,  the  projection  of  the 
vertical  axis  of  the  instrument  upon  the  plane  HZS\ 


INSTRUMENTS.  113 

and  let  rf  denote  the  reading  of  the  vertical  circle  when 
the  line  of  sight  is  directed  toward  5.  After  reading  rf 
let  the  instrument  be  turned  about  the  vertical  axis 
through  an  angle  of  180°,  bringing  the  line  of  sight  into  the 
position  HSf,  and  then  let  the  telescope  be  turned  about 
the  horizontal  axis  until  the  line  of  sight  again  points 
at  the  object  5,  and  let  r"  be  the  reading  of  the  vertical 
circle  in  this  position.  If  the  circle  is  numbered  in  quad- 
rants, as  is  very  common  in  small  instruments,  r'  and  r" 
will  be  approximately  the  same  number  but  with  a 
graduation  extending  from  o°  to  360°,  as  is  here  sup- 
posed, they  will  be  widely  different.  From  the  figure, 
the  angle  SHS'  is  measured  by  the  difference  of  these 
circle  readings,  r'  —  r",  and  since  VHS  =  VHS',  we  have 
for  the  angular  distance  of  the  point  5  from  V  the 
equation, 

r").  (89) 


When  two  pointings  of  the  telescope  are  made  as 
above,  the  instrument  is  said  to  be  reversed  between  them, 
and  it  is  customary  to  designate  its  two  positions  as 
Circle  Right  and  Circle  Left,  respectively,  the  reference 
being  to  the  vertical  circle  of  the  instrument,  which 
faces  to  the  observer's  right  in  the  one  position  and  to 
his  left  in  the  other.  The  student  should  note  that  the 
angle  z"  is  here  determined  quite  independently  of  the 
adjustment  of  the  verniers,  which  may  be  intended  to 
read  altitudes,  zenith  distances,  or  anything  else,  since 
the  reversal  eliminates  all  question  of  adjustment  from 
the  difference  r'  —  r"  ,  and  is  made  for  this  purpose. 


1U  FIELD  ASTRONOMY. 

The  true  zenith  distance  of  5  is,  however,  not  z"  but 
the  angle, 


and  b"  may  be  determined,  as  in  §  42,  from  readings 
of  the  spirit-level,  LL,  attached  to  the  instrument  in 
such  a  way  that  its  plane  is  parallel  to  the  line  of  sight, 
HS.  Such  a  level,  i.e.,  one  whose  tube  is  perpendicular 
to  the  horizontal  axis  of  the  instrument,  will  be  called 
the  altitude  level  of  the  instrument. 

A  convenient  method  of  taking  into  account  the 
readings  of  the  level-bubble  by  applying  them  directly 
to  the  circle  readings  instead  of  to  the  measured  angles, 
is  as  follows:  Let  n0  represent  the  reading  of  that  point 
of  the  level  scale  through  which  passes  that  radius  of 
the  level  which  is  parallel  to  the  vertical  axis,  HV,  CnQ 
in  the  figure,  and  let  n  denote  the  position  of  the  middle 
of  the  bubble  corresponding  to  the  circle  reading  /; 
i.e.,  since  the  bubble  always  stands  at  the  highest  point 
of  its  tube,  n  is  the  point  exactly  above  the  centre  of 
curvature,  C.  It  is  evident  from  the  figure  that 

b"  =  (n-nQ)2d  =  (a  +  b-  2n())d,  (90) 

where  d  is  the  value  of  half  a  level  division,  and  a  and  b 
are  the  actual  scale  readings  of  the  ends  of  the  bubble. 

If  the  instrument  had  been,  from  the  first,  perfectly 
levelled  we  should  not  have  obtained  r'  as  the  reading 
to  the  point  5,  but  in  place  of  rh  a  number  either  greater 
or  less  than  it  by  the  amount  b"  ;  and  if,  therefore,  we 
apply  to  /  and  r"  level  corrections  determined  by  the 
equation  above  given  for  6",  we  shall  reduce  the  read- 


PLATE  III. 


A  German  Universal  Instrument.     Length  of  Horizontal  Axis 
12  inches.     Approximate  Cost  3400. 

[Tofacep.  114.] 


INSTRUMENTS.  115 

ings  to  what  they  would  have  been  for  a  perfectly  levelled 
instrument,  and  therefore  obtain  the  zenith  distance 
of  5  immediately  from  the  half  difference  of  the  corrected 
readings.  Since  any  constant  term  which  appears  in  the 
level  correction  will  be  eliminated  from  this  difference  of 
the  corrected  readings,  r'-r",  we  may  substitute  in 
Equation  90,  in  place  of  2nQ,  any  constant  number  what- 
ever, e.g.,  zero,  but  it  is  usually  convenient  to  take  as 
this  number  5,  the  total  number  of  divisions  included  in 
the  level  scale,  since  in  the  long  run  this  will  make  the 
level  corrections  small.  Making  this  substitution,  we 
have  finally, 

Level  Correction  =  ±  (a  +  b  —  S)d,  (91) 

where  the  ambiguous  sign  depends  upon  the  direction 
in  which  the  numbers  increase  along  the  level  scale, 
and  may  be  determined,  once  for  all,  for  a  given  instru- 
ment as  follows:  Two  readings  of  the  vertical  circle 
of  a  certain  instrument  were  taken  to  the  same  object, 
but  with  the  instrument  thrown  out  of  level  in  such  a 
way  that  the  bubble  stood  at  quite  different  parts  of 
the  scale  in  the  two  observations ;  e.g. : 

Observation.  Bubble.  Circle.  Level  Corr.       Corrected  r. 

First 2*0     25.8     91°  9'     8'         +i8".7      91   9'  26". 7 

Second 7.9     31.9     91  9    40         —12    .5      91  9    27   .5 

The  numerical  values  of  the  quantities  above  marked  Level  Corr. 
were  computed  from  Equation (91)  with  an  assumed  value  of  d  =  2".6, 
and  since  the  effect  of  these  corrections  must  be  to  bring  the  corrected 
circle  readings  into  agreement,  it  is  evident  that  the  -f  sign  must  be 
used  for  the  first  observation  and  the  —  sign  for  the  second.  The 
whole  number  of  divisions  in  the  level  scale  being  35,  the  formula  for 
this  instrument  becomes, 

6"  =  +2".6  [3- 


116  FIELD  ASTRONOMY. 

A  similar  formula  may  be  obtained  for  every  instrument,  and  a  table 
should  be  constructed  from  it,  which  with  the  argument  a  +b  will 
show  the  value  of  b"  for  any  given  position  of  the  bubble.  Part  of 
such  a  table  is  given  below  ,4 and  from  it,  the  level  correction  correspond- 
ing to  any  ordinary  position  of  the? '"bubble  may  be  determined  by 
inspection. 

a  +  b  b"  a  +  b 

30  +  i3".o  —          40 

31  +10    .4  —          39 

32  +    7    -8-          38 

33  +5-2-          37 

34  +    2  .6-  36 

35  +o    .o-          35 

In  the  second  observation  given  above  we  have  0  +  6  =  7.9  +  31.9=39.8, 
and  corresponding  to  this  number  we  find,  by  interpolation  from  the 
table,  6"  =  -i2"/5. 

The  level  formulae  thus  derived  show  that  if  the 
bubble  be  brought  to  the  same  place  in  the  tube,  same 
values  of  a  and  6,  both  Circle  R.  and  Circle  L.,  the  level 
correction  will  be  eliminated  from  the  difference  r'  —  r"  y 
and  may  therefore  be  neglected.  To  obtain  the  maxi- 
mum precision  however,  the  level  should  always  be 
read  and  a  correction  applied  to  each  circle  reading,  but 
even  when  this  is  done  it  is  good  practice  to  touch  up  the 
levelling  screws  after  each  reversal  and  bring  the  bubble 
back  as  near  as  may  be  to  its  first  position,  without,  how- 
ever, spending  too  much  time  in  obtaining  an  accurate 
agreement. 

In  some  instruments  the  level  and  verniers  are  at- 
tached to  a  frame  (alidade)  which  admits  of  rotation 
about  the  horizontal  axis  without  disturbing  the  direc- 
tion of  the  line  of  sight.  For  such  an  instrument  it 
may  be  shown  that  if,  before  reading  the  vernier,  the 
frame  be  turned  until  the  bubble  stands  at  the  middle 
of  the  scale,  the  resulting  vernier  readings  are  equiva- 


INSTRUMENTS. 


117 


lent  to  the  corrected  circle  readings  derived  above,  and 
therefore  require  no  further  correction  for  level  error. 
This  mechanical  device,  although  convenient  for  some 
purposes,  is  of  inferior  accuracy. 

49.  Effect  of  Errors  of  Adjustment.  —  A  geometrical 
investigation  similar  to  the  above  may  be  made  to  show 
the  effect  of  each  source  of  instrumental  error,  but  we 
shall  find  it  more  convenient  to  develop  the  combined 
effect  of  these  errors  through  an  analysis  based  upon 
Fig.  10,  which  represents  a  part  of  the  celestial  sphere, 


FIG.  io.— Theory  of  the  Theodolite. 

where  Z  is  the  zenith,  V  is  the  point  in  which  the  vertical 
axis  of  the  instrument,  when  produced,  cuts  the  sphere, 
H  is  the  point  of  the  sphere  determined  by  the  prolonga- 
tion of  the  horizontal  axis,  and  5  is  a  star  or  other  object 
whose  azimuth  and  altitude  are  to  be  determined  from 
readings  of  the  horizontal  and  vertical  circles  of  the 
instrument.  The  angles  measured  by  means  of  these 


118  FIELD  ASTRONOMY. 

circles  lie  in  the  planes  of  the  circles,  but  just  as  the 
azimuth  of  a  point  is  measured  either  by  an  arc  of  the 
horizon  or  by  the  corresponding  spherical  angle  at  the 

*     ,  •* 

zenith,  so  the  data  furnished  by  the  vernier  readings 
may  be  regarded  as  spherical  angles  having  their  ver- 
tices respectively  at  V  and  H.  Thus  if  r  represent  the 
reading  of  the  vertical  circle  when  the  line  of  sight  is 
directed  toward  S,  and  rQ  is  the  reading  when  this  line 
is  directed  toward  some  point  in  the  arc  HV,  the  differ- 
ence, r  —  r0,  measures  the  spherical  angle  VHS.  Simi- 
larly, for  the  horizontal  circle,  by  rotating  the  instrument 
about  its  vertical  axis,  H  may  be  moved  from  its  present 
position,  corresponding  to  the  reading  R,  into  a  new 
position  falling  upon  the  arc  VM,  and  if  Rl  be  the  circle 
reading  in  this  position,  we  shall  find  that  R  —  R1  equals 
the  spherical  angle  HVM.  From  these  spherical  angles, 
determined  by  the  circle  readings,  it  is  required  to  find 
the  true  direction,  MZS  =  A'y  and  the  true  zenith  dis- 
tance, ZS  =  2,  of  the  star  5. 

It  is  evident  from  the  figure  that  the  arc  VH  =  90°  —  i, 
measures  the  angle  between  the  vertical  and  the  horizontal 
axis  of  the  instrument,  and  that  i  is  therefore  the  error 
of  adjustment  of  the  axes,  corresponding  to  Condition  i, 
§  47.  Similarly,  HS  =  90°  +  c  measures  the  angle  be- 
tween the  horizontal  axis  and  the  line  of  sight,  and  c  is 
the  error  in  the  adjustment  corresponding  to  Condition  2. 
Also,  VZ  =  f  is  the  error  of  level  of  the  instrument, 
i.e.,  deviation  of  the  vertical  axis  from  the  true  vertical, 
corresponding  to  Condition  3.  The  arc  HZ  =  Qo°-b 
measures  the  angle  that  the  horizontal  axis  makes  with 


INSTRUMENTS.  119 

the  true  vertical,  and  b  is  therefore  the  level  error  of  this 
axis.  Note  that  as  the  instrument  is  turned  into  differ- 
ent positions  by  rotation  about  the  axes  V  and  //,  the 
quantities  ?-,  i,  and  c  remain  unchanged  and  are  there- 
fore called  instrumental  constants,  since  they  define 
the  condition  of  the  instrument  with  respect  to  its 
several  adjustments.  The  level  error,  b,  is  sometimes 
included  among  these  constants,  but  is  not  strictly  one 
of  them,  since  its  value  changes  as  the  instrument  is 
turned  in  azimuth. 

We  shall  suppose  the  instrument  to  be  so  well  ad- 
justed that  none  of  the  instrumental  constants  exceeds 
2',  and  H  will  then  be  so  near  the  pole  of  the  great  cir- 
cle VZM  that  we  may  assume  without  sensible  error 
HVM=HZM  and,  replacing  these  quantities  by  their 
equivalents,  obtain 


or 

A'=R-(Rl  +  9o°)-w.  (92) 

The  azimuth  of  5,  reckoned  from  the  true  meridian 
instead  of  from  the  arc  VM,  differs  from  A'  only  by 
the  substitution  of  another  constant,  the  index  correc- 
tion of  the  horizontal  circle,  in  place  of  Rl  +  90°  ;  and  as 
this  index  correction  must  in  any  case  be  separately 
determined  (see  §  38),  we  may  replace  the  constant  term 
R^  +  90°  by  R0,  the  index  correction  referred  to  the  true 
meridian,  and  we  shall  then  have  for  the  true  azimuth  of  S, 

A-R-Rt-w.  (93) 


120  FIELD  ASTRONOMY. 

The  auxiliary  quantity  w  has  thus  far  been  denned 
only  by  means  of  Fig.  10,  where  the  spherical  angle  HZS 
is  labelled  90°+  w.  To  .determine  the  value  of  w  in  terms 
of  the  instrumental  constants;  we  have  from  the  triangle 
HZS  by  means  of  Equations  4,  the  relation, 

—  sin  c  =  sin  b  cos  2  —  cos  b  sin  z  sin  w, 
which,  since  b  and  c  are  small  quantities,  is  equivalent  to 


(94) 


or,  replacing  the  zenith  distance,  z,  by  the  star's  alti- 
tude, h, 

w  =  c  sec  h  +  b  tan  h.  (94*) 

Since  neither  i  nor  7-  enters  into  this  equation,  the  effect 
of  these  errors  must  be  taken  into  account  through  6, 
the  inclination  of  the  horizontal  axis.  This  is  to  be  deter- 
mined with  a  spirit-level,  and  each  circle  reading,  R, 
must  be  corrected  for  the  particular  inclination  of  the 
axis  that  corresponds  to  R.  The  factor  tank  becomes 
zero  for  an  object  in  the  horizon,  and  for  this  special  case 
the  effect  upon  the  azimuth  readings  of  an  error  of  level 
is  zero.  On  the  other  hand,  when  the  object  to  be 
observed  is  at  a  considerable  elevation,  e.g.,  the  Pole 
Star  in  an  azimuth  determination,  the  factor,  tan  h, 
becomes  large  and  the  effect  of  level  error  is  magnified. 
It  is  in  fact  one  of  the  chief  sources  of  error  in  such  deter- 
minations. 

50.  Determination  of  Errors  of  Adjustment.  —  The  error 
above  represented  by  c  is  called  the   collimation,  and 


INSTRUMENTS.  121 

its  effect  is  usually  to  be  eliminated  through  a  reversal 
of  the  instrument.  Since  the  angular  distance  of  5  from 
one  end  of  the  horizontal  axis  is  90°  +  c,  its  distance 
from  the  other  end  must  be  90°  —  c,  and  as  in  the  rever- 
sal these  ends  change  places  the  effect  of  c  must  have 
one  sign  Circle  R.,  and  the  opposite  sign  Circle  L.,  and 
will  therefore  be  eliminated  from  the  mean  of  observa- 
tions taken  in  both  positions. 

In  precisely  the  same  way  it  may  be  shown  that  the 
effect  of  i,  error  of  adjustment  of  the  axes,  is  eliminated 
from  the  mean  of  observations  taken  in  the  two  posi- 
tions, and  wherever  any  considerable  precision  is  required 
in  azimuth  observations  or  in  the  measurement  of  hori- 
zontal angles,  the  observer  should  not  fail  to  make  an 
equal  number  of  pointings  in  each  position  of  the  instru- 
ment to  secure  this  elimination  of  errors. 

In  the  triangle  HVZ  the  angle  HVZ  =  HZM  is  very 
nearly  equal  to  90°  +  Af ,  and  assuming  this  equality  we 
•find  from  this  triangle, 

sin  b  =  sin  i  cos  f  —  cos  i  sin  7-  sin  A',  (95) 

which  is  equivalent  to, 

b=i-r$mAf.  (96) 

The  quantity  7-  sin  A' ',  which  we  shall  represent  here- 
after by  the  symbol  6',  and  which  corresponds  to  the 
arc  ZI  of  Fig.  10,  is  that  component  of  the  level  error 
of  the  vertical  axis,  7-,  which  lies  at  right  angles  to  the 
line  of  sight  and  which  may  therefore  be  determined 
from  the  readings  of  a  level  parallel  to  the  horizontal 


122  FIELD  ASTRONOMY. 

axis  of  the  instrument.  Such  a  level  is  called  the  azi- 
muth level,  and  if  resting  upon  the  axis  and  capable  of 
reversal  (striding-level),  it  is  most  conveniently  used 
to  determine  the  level  error  of :  this  axis,  b.  If  fastened 
to  the  frame  of  the  instrument  and  incapable  of  reversal, 
it  may  be  used  to  determine,  from  bubble  readings  taken 
Circle  R.  and  Circle  L.,  the  value  of  b'  for  the  vertical 
axis,  and  corresponding  to  these  two  cases  we  shall  have 
the  following  expressions  of  the  level  corrections  to  be 
applied  to  readings  of  the  horizontal  circle : 

Striding-level,    -  b  tan  h.     Both  Circle  R.  and  Circle  L. 
Fixed  Level,      —  b'  tan  h.    Mean  of  Circle  R.  and  L. 

If,  as  is  usual,  the  graduation  of  the  circle  increases  from 
left  to  right,  b  and  bf  are  to  be  considered  essentially 
positive  when  the  high  end  of  the  horizontal  axis  has  an 
azimuth  90°  greater  than  the  object  5. 

The  student  should  not  fail  to  note  in  connection  with 
the  use  of  a  fixed  azimuth  level  that  if  the  bubble  is 
brought  to  the  same  scale  reading,  Circle  R.  and  Circle  L., 
b'  will  be  zero  and  the  level  error  will  be  eliminated  from 
the  mean  result. 

A  reversal  furnishes  a  convenient  method  for  deter- 
mining or  adjusting  the  collimation.  For  this  purpose 
let  R'  and  R"  be  readings  of  the  horizontal  circle  corre- 
sponding to  observations  of  a  fixed  mark  in  or  very  near 
the  horizon,  made  in  the  two  positions  of  the  instrument ; 
then,  from  Equations  92  and  94, 

ic-R-K'.  (97) 


INSTRUMENTS.  123 

To  determine  the  error  of  adjustment  of  the  axes,  i,  let 
the  inclinations  of  the  horizontal  axis,  bv  62,  be  meas- 
ured in  two  positions  of  the  instrument  differing  180° 
in  azimuth,  i.e.,  when  Vernier  A  reads  o°  and  when  it 
reads  180°.  We  shall  then  have,  from  Equation  96, 

b1=i—^  sin  A', 


sin  A', 

sin  (Ar  +  180°)  =i+  r  sin  A', 


from  which  we  obtain  immediately 

2*  =  ^  +  62.  (98) 

If  we  call  the  inclinations  blt  b2  positive  when  the  circle 
end  of  the  axis  is  too  high,  a  positive  value  of  i  will  indi- 
cate that  the  same  end  is  too  high,  i.e.,  it  makes  too 
small  an  angle  with  the  upward  extension  of  the  vertical 
axis. 

The  value  of  7-,  which  will  seldom  be  required,  may 
be  found  from  four  values  of  6  determined  at  intervals 
of  90°  in  azimuth. 

51.  Additional  Theorems.  —  By  an  analysis  similar  to 
that  employed  above,  it  may  be  shown  from  the  trian- 
gles HSZ,  HZV,  of  Fig.  10,  that  the  errors  b,  c,  and  i  have 
no  appreciable  influence  upon  observations  of  altitude 
or  zenith  distance.  Indeed,  it  may  be  seen  without 
formal  analysis  that  when  c,  6,  and  i  are  small  quantities, 
H  is  so  nearly  the  pole  of  the  circles  ZS,  VS,  that  these 
arcs  are  measured  by  the  corresponding  angles  at  H, 
i.e.,  by  the  readings  of  the  vertical  circle  unconnected 
for  instrumental  error.  Since  the  error  corresponding 
to  Y  is  taken  into  account  in  the  approximate  analysis 


FIELD  ASTRONOMY. 

of  §  48,  we  may  adopt  as  definitive  the  results  there 
obtained.  The  correction  b"  there  determined  is  the 
arc  VI  of  Fig.  10,  i.e.,  it  is  the;  projection  of  ?  upon  the 
line  of  sight,  VS. 

The  demonstration  of  the  following  theorems,  which 
are  of  some  consequence  in  the  use  of  a  theodolite,  is 
left  to  the  student. 

i.  If,  as  is  quite  common  in  engineer's  transits,  the 
vertical  circle  is  graduated  into  quadrants  instead  of 
from  o°  to  360°,  observations  of  altitude  should  be  made 
in  the  way  already  indicated,  but  in  their  reduction 
we  shall  have,  in  place  of  the  formula  for  z",  the  substi- 
tute, 

r"),  (99) 


i.e.,  the  mean  of  the  readings  gives  directly  the  instru- 
mental altitude. 

2.  The  altitude  level  of  such  an  instrument  usually 
has  the  zero  of  its  scale  placed  at  the  middle  of  the  tube, 
and  when  such  is    the  case  readings  of  that  end  of  the 
bubble  nearest  the  objective  end  of  the  telescope  should 
be  marked  o,  and  those  of  the  end  nearest  the  eyepiece 
should  be  called  e\  the  formula  for  level  correction  then 
becomes, 

b"  =  (o-e)d.  ,  (100) 

3.  A  theodolite  maybe  reversed  by  lifting  the  tele- 
scope from  its  supports,  turning  the  axis  end  for  end, 
and  replacing  it  in  the  wyes  in  the  changed    position. 
This  mode  of  reversal  eliminates  errors  of  level  and  colli- 
mation  quite  as  well  as  does  the  one  above   described, 


INSTRUMENTS.  125 

and  also  eliminates  the  inequality  of  pivots  from  the 
determination  of  b.  It  is  therefore  to  be  preferred 
when  it  can  be  conveniently  practised. 

52.  Errors  Arising  from  the  Circle  Readings.  —  Numerous 
errors  of  a  class  not  considered  above,  creep  into  the 
results  of  observation  through  the  circle  readings,  which 
may  be  vitiated  in  greater  or  less  degree  by: 

(a)  Defective  graduation  of  the  circle  itself. 

(b)  The  plane  of  the  circle  not  being  normal  to  the 

rotation  axis. 

(c)  The  circle  not  being  truly  centred  upon  the  axis. 

(d)  The  spaces  on  the  vernier  being  too  large  or  too 

small  relative  to  those  on  the  circle. 

(e)  Error  of  focussing  (runs)  in  the  reading  micro- 

scopes, 
etc.  etc.  etc. 

The  detailed  study  of  these  sources  of  error  lies  be- 
yond the  scope  of  the  present  work,  but  we  note  that  in 
great  part  their  effects  may  be  eliminated  by  taking  the 
mean  of  a  considerable  number  of  observations  in  which 
the  circle  readings  are  symmetrically  distributed  through- 
out the  whole  360°  of  the  graduation.  Thus  if  an  angle 
of  120°  between  objects  A  and  B  is  measured  three  times 
and  the  circle  turned  120°  after  each  measurement  so 
as  to  obtain  the  following  system  of  readings : 

To  A.  To  B.  B-A. 

Observation  i o°  o'  o"  120°  o'  o"   120°  o'  o" 

2 I2O   O   O   240   O   O    120   O   O 

3 240  o  o  360  o  o   120  o  o 


126  FIELD  ASTRONOMY. 

whatever  graduation  errors  may  affect  the  particular 
reading  120°  o'  oo"  will  be  eliminated  from  the  mean 
value  of  B  —  A,  since  this  reading; enters  into  that  mean 
once  with  a  plus  sign  and  once' with  a  minus  sign.  If 
the  required  angle  is  small,  e.g.,  i°,  it  will  not  be  con- 
venient to  carry  out  the  above  programme  of  reading 
around  the  entire  circle,  but  the  elimination  of  errors 
may  still  be  made  by  shifting  the  circle  so  that  the 
readings  to  object  A  may  be  symmetrically  distributed 
through  the  entire  circumference,  e.g.,  every  60°  or 
every  30°.  For  an  instrument  provided  with  two  ver- 
niers or  microscopes  it  will  suffice  to  distribute  the  read- 
ings of  each  vernier  over  an  arc  of  180°. 

53.  The  Method  of  Repetitions.  —  A  peculiar  method 
of  measuring  horizontal  angles  may  be  adopted  with 
advantage  if,  as  is  often  the  case,  the  instrument  is 
provided  with  two  motions  in  azimuth  called,  respectively, 
upper  and  lower,  one  of  which  produces  a  change  in  the 
vernier  readings,  while  in  the  other,  verniers  and  circle 
remain  firmly  clamped  together  and  turn  simultaneously, 
without  change  in  the  circle  reading.  Reverting  to 
§  52,  we  may  note  that  the  circle  readings  120°,  240°, 
there  recorded,  are  quite  unnecessary  since,  if  the  first 
reading,  o°,  be  subtracted  from  the  last  one,  360°,  and 
the  result  divided  by  3,  we  shall  have  as  the  value  of 
the  angle  120°  o'  o",  which  is  precisely  the  same  as 
the  mean  of  the  three  values  of  B  —  A,  and  is  all  that 
that  mean  can  furnish. 

This  process  is  called  the  method  of  repetitions  and 
consists,  essentially,  in  making  a  series  of  pointings  upon 


INSTRUMENTS.  127 

two  objects  between  which  an  angle  is  to  be  measured, 
turning  always  from  A  to  B  upon  the  upper  motion  of 
the  instrument  and  from  B  to  A  upon  the  lower  motion, 
so  that  the  vernier  reading  in  the  latter  turning  is  not 
changed.  A  series  of  such  pointings  is  called  a  set  and 
the  verniers  need  be  read  only  for  the  first  and  last 
pointings  of  the  set.  If  the  initial  and  final  readings  be 
represented  by  Rr  and  R",  and  n  be  the  number  of  point- 
ings to  each  object  contained  in  the  set,  we  shall  have,  as 
shown  above, 

.      ,        R'-R" 

Angle  =  — ^— •  (101) 

It  is  often  advantageous  to  reverse  the  instrument  at 
the  middle  of  a  set,  turning  on  the  lower  motion,  and  thus 
secure  an  additional  elimination  of  instrumental  errors. 

The  advantages  of  the  method  of  repetitions  are  a 
saving  of  labor  through  the  diminished  number  of  vernier 
readings  and,  where  the  verniers  are  comparatively 
coarse,  an  increase  of  accuracy  through  the  introduction 
of  the  divisor  n  into  the  value  of  the  angle.  The  pre- 
cision of  a  small  instrument,  such  as  an  engineer's  transit, 
may  be  considerably  increased  in  this  way,  but  for  the 
larger  instruments,  provided  with  micrometer  micro- 
scopes, experience  shows  that  the  best  results  are  to  be 
obtained  by  reading  the  microscopes  after  every  pointing. 

Where  a  horizontal  angle  between  objects  at  very 
different  altitudes  is  to  be  measured  by  the  method  of 
repetitions,  as  in  an  azimuth  determination,  an  addi- 
tional source  of  error  requires  careful  attention,  viz.,  the 
effect  of  a  lack  of  parallelism  between  the  axes  corre- 


128  FIELD  ASTRONOMY. 

spending  to  the  upper  and  lower  motions  of  the  instru- 
ment. To  eliminate  this  error  we  proceed  in  the  follow- 
ing manner:  The  axis  of  the  lower  motion  should  be 
made  as  nearly  vertical  as  possible,  and  whatever  may 
be  the  error  of  the  upper  axis  it  will  produce  no  effect 
upon  the  final  result  if  the  number  of  repetitions  is  so 
chosen  that  the  set  extends  through  360° ;  for  in  the 
successive  turnings  about  the  lower  motion  the  upper 
axis  has  been  made  to  describe  a  complete  cone  about 
the  lower  axis,  and  any  error  which  may  have  been  caused 
by  a  deflection  to  the  east  in  one  part  of  the  set  is  bal- 
anced by  the  opposite  error,  caused  by  a  deflection  to 
the  west,  in  another  part,  etc.  If  the  angle  to  be  meas- 
ured is  so  small  that  the  set  cannot  be  made  to  extend 
through  360°,  the  following  observing  programme  will 
also  eliminate  the  error  of  the  axis :  Measure  a  set  of  any 
desired  number  of  repetitions.  When  it  is  completed 
leave  the  instrument  clamped  at  the  last  vernier  read- 
ing, reverse  about  the  lower  motion  and  repeat  the  set  in 
the  opposite  direction,  i.e.,  beginning  with  the  object 
last  sighted  upon  and  with  approximately  the  vernier 
reading  last  obtained. 

The  level  correction  to  the  circle  readings  should  be 
derived  in  the  ordinary  way,  §  50,  from  readings  of  a 
level  taken  when  the  instrument  is  reversed  about  the 
lower  axis. 

54.  Precepts  for  the  Use  of  a  Theodolite.  —  The  ex- 
perience of  the  principal  geodetic  surveys  indicates  that 
the  following  precepts  should  be  observed  in  all  precise 
work  with  a  theodolite : 


INSTRUMENTS.  129 

(1)  An  equal  number  of  measurements   should  be 
made  in  each  position  of  the  instrument,  Circle  R.  and 
Circle  L. 

(2)  An  equal  number  should  be  taken  in  each  direc- 
tion, i.e.,  the  line  of  sight  turned  from  right  to  left  and 
from  left  to  right. 

(3)  The  position  of  the  circle  should  be  so  shifted  from 
time  to  time  that  the  readings  to  each  object  are  sym- 
metrically distributed  throughout  the  360°. 

(4)  The  observations  should  be  made  as  rapidly  as 
the  observer  can  work  without  undue  haste. 

55.  The  Sextant. — A  sextant  consists  essentially  of 
two  mirrors  and  a  graduated  arc  of  a  circle,  about  60°, 
for  measuring  the  angle  between  the  planes  of  the  mir- 
rors. The  peculiar  value  of  the  instrument  lies  in  the 
fact  that  it  is  light  and  portable,  requires  no  fixed  support, 
and  may  therefore  be  used  for  the  measurement  of 
angles  at  sea  as  well  as  on  shore,  and  in  any  plane,  ver- 
tical, horizontal,  or  inclined.  For  the  purpose  of  de- 
scription and  analysis  we  suppose  the  sextant  to  be 
placed  upon  a  table,  with  the  plane  of  its  arc  horizontal, 
and  we  shall  use  the  terms  altitude,  azimuth,  etc.,  with 
reference  to  this  special  position  of  the  instrument.  The 
conclusions  drawn  from  this  consideration  of  the  in- 
strument apply  equally  when  it  is  used  in  any  other  plane. 

The  essential  parts  of  a  sextant  are  indicated  in 
Fig.  ii  which  should  be  compared  with  Plate  IV.  At 
the  centre  of  the  arc  is  a  vertical  axis  carrying  a 
vernier-arm,  V,  and  also  supporting  one  of  the  mirrors 
called  the  index-glass,  7,  whose  plane  is  vertical,  passes 


130  FIELD  ASTRONOMY. 

nearly  through  the  axis  and  rotates  with  the  vernier- 
arm  as  the  latter  is  turned  in  azimuth.  At  one  side  of 
the  sextant  frame  is  the  other  mirror,  H,  called  the  hori- 
zon-glass, with  its  plane  vertical  and  fixed  parallel  to 
that  radius  of  the  graduated  arc  which  is  numbered  o°. 
Only  the  lower  half  of  the  horizon-glass  is  silvered,  the 
upper  half  is  left  transparent.  A  telescope,  T,  is  mounted 
on  the  side  of  the  frame  opposite  to  the  horizon-glass 
and  has  its  line  of  sight  directed  toward  the  latter. 
From  Fig.  n  it  may  be  seen  that  an  observer  looking 


FIG.  n. — Elements  of  a  Sextant. 

into  the  telescope  and  through  the  unsilvered  upper  half 
of  the  horizon -glass  will  see  that  part  of  the  horizon 
toward  which  the  telescope  is  directed,  and  will  also  see 
superposed  upon  it  a  view  of  another  part  of  the  horizon 
reflected  from  the  index -glass  to  the  silvered  half  of  the 
horizon-glass,  and  from  this  again  reflected  into  the  tele- 
scope. This  part  of  the  horizon  is  said  to  be  seen 


I 


*  o 

>  2 


INSTRUMENTS.  131 

reflected,  while  the  part  seen  through  the  horizon-glass  is 
observed  direct.  Any  reflected  image  which  is  super- 
posed upon  a  direct  image  is  said  to  be  in  contact  with 
the  latter,  and  we  shall  represent  these  images  as  seen 
in  the  telescope,  by  /  and  H  respectively. 

By  turning  the  index -glass  in  azimuth,  different  parts 
of  the  horizon  may  be  reflected  into  the  telescope,  and 
since  the  rays  of  light  incident  upon  and  reflected  from 
the  mirror  make  equal  angles  with  its  surface,  it  is 
apparent  that  for  every  i°  that  the  mirror  is  turned,  the 
azimuth  of  the  point  reflected  into  the  telescope  will 
be  changed  by  2°.  There  may  be  found  by  trial  a  set- 
ting of  the  index -glass  at  which  both  a  direct  and  a  re- 
flected image  of  the  same  object  may  be  seen  simul- 
taneously and  may  be  made  to  pass  one  over  the  other 
as  the  vernier-arm  is  slightly  turned.  Let  R0  denote 
the  vernier  reading  when  these  images  are  brought  into 
contact,  and  let  R  be  the  reading  at  which  any  other 
object,  /,  is  brought  into  contact  with  the  H  just  ob- 
served; then  it  appears  from  the  above  that  the  differ- 
ence of  azimuth  between  /  and  H  is  twice  the  angle 
included  between  R0  and  R.  On  account  of  this  mul- 
tiplier, 2,  each  half -degree  of  the  sextant  arc  is  numbered 
as  if  it  were  a  whole  degree,  and  we  have,  therefore,  for 

the  difference  of  azimuth, 

i 

H-I=R-R0.  (102) 

The  —  R0  which  appears  in  this  equation  is  called  the 
index  correction,  and  it  should  be  observed  that,  owing 
to  the  angle  subtended  at  the  object  H  by  the  space 


132  FIELD  ASTRONOMY. 

separating  the  index  and  horizon  glasses,  the  reading  RQ 
will  depend  upon  the  distance  of  H  from  the  instrument. 
If  it  is  near  at  hand,'  less  than  three  miles,  R0  should  be 
determined  as  above  and  the  axis  of  rotation  of  the 
index -glass  should  be  centred  over  the  point  at  which  it 
is  desired  the  vertex  of  the  measured  angle  should  fall. 
If  the  objects  are  very  remote,  all  question  of  the  exact 
position  of  the  vertex  is  eliminated,  and  a  mode  of  deter- 
mining the  index  correction  given  hereafter  will  be  found 
more  convenient  than  the  above. 

It  may  now  be  seen  that  the  following  conditions 
must  be  satisfied  in  order  that  the  Equation  102,  given 
above,  shall  furnish  the  true  value  of  the  angle  between 
H  and  /: 

(a)  The  rotation  axis  and  the  plane  of  the  index -glass 
must  be  perpendicular  to  the  plane  of  the  graduated 
arc.     If   they    are   not   perpendicular,  this    arc    cannot 
accurately  measure  the  amount  of  rotation  of  the  mirror. 

(b)  The  horizon-glass  must  be  perpendicular  to  the 
plane  of  the  arc.     If  it  is  not  perpendicular,  the  direct 
and  reflected  images  of  H  cannot  be  brought  into  con- 
tact,.but  one  will  pass  above  or  below  the  other  as  the 
vernier -arm  is  turned. 

(c)  The  objects  H  and  /  must  lie  in  the  sextant  hori- 
zon, for  otherwise  the  difference  of  their  azimuths  would 
not   be    the    true    angle    between    them.     The    sextant 
horizon  must  here  be  understood  to  mean  the  plane  of 
the  graduated  arc,  and  this  condition  will  be  satisfied 
if  the  sextant  is  so  held  during  the  observation  that  this 
plane  passes  through  the  objects  H  and  /. 


INSTRUMENTS  133 

56.  Adjustments  of  the  Sextant. — (A)  The  Index-glass. 
— Take  the  telescope  out  from  its  support,  set  it  on 
end  at  any  part  of  the  arc,  and  turn  the  index -glaso 
until  its  plane  passes  a  little  to  one  side  of  the  telescope. 
By  holding  the  eye  a  little  to  the  right  of  the  line  joining 
the  index -glass  to  the  telescope  a  reflected  image  of  the 
telescope  may  be  seen  simultaneously  with  a  direct  view 
of  it,  and  these  two  images  should  be  parallel,  provided 
the  telescope  stands  normal  to  the  plane  of  the  arc. 
Any  error  in  this  last  condition  may  be  eliminated  by 
turning  the  telescope  180°  about  its  own  axis  and  repeat- 
ing the  test.  No  adjusting-screws  are  provided  for  the 
index-glass,  but  it  may  be  adjusted,  if  necessary,  by  re- 
moving it  from  its  frame  and  filing  down  the  bearing- 
points  against  which  it  is  held. 

(B)  The  Horizon-glass. — Bring    the    direct    and    re- 
flected images  of  a  distant  object  into  contact  if  possible. 
If  this  cannot  be  done,  bring  them  near  together  and 
tilt  the  horizon -glass  by  means    of  its    adjusting  screws 
until  by  turning  the  vernier-arm  the  images  can  be  made 
to  coincide. 

(C)  The  Telescope. — To  enable  the  observer  to  make 
the  plane  of  the  sextant  pass  through  the  objects  H 
and  /  it  is  customary  to  place  in  the  eyepiece   of  the 
telescope  a  pair  of  coarse  threads  which  should  be  set 
parallel  to  the  plane  of  the  sextant.     By  means  of  its 
adjusting  screws  the  telescope    should  be  tilted  up  or 
down  until  the  line  of  sight  passing  midway  between 
these  threads  is  parallel  to  the  plane  of  the  sextant.    If 
the  objects  H  and  7  are  brought  midway  between  these 


134  FIELD  ASTRONOMY. 

threads  when  contact  between  them  is  made,  they  will 
lie  in  the  plane  of  the  sextant  as  required.  To  deter- 
mine if  the  telescope  is  properly  tilted,  select  two  well- 
defined  objects  about  120°  apart,  and  bring  them  into 
contact  when  the  sextant  is  so  held  that  they  are  both 
seen  in  the  upper  part  of  the  field  of  view.  Then  shift 
the  position  of  the  sextant  plane  so  as  to  bring  the 
objects  to  the  lower  part  of  the  field  and  note  whether 
they  remain  in  contact  or  appear  separated ;  if  they  are 
appreciably  separated  the  telescope  requires  further 
adjustment. 

57.  Outstanding  Errors  of  the  Sextant. — The  methods 
of  adjustment  above  described  are  only  approximate,  and 
the  readings  of  the  instrument  will  be  affected  by  what- 
ever error  remains  in  the  adjustment.  In  general  the 
effect  of  these  errors  will  be  small  for  small  angles,  but 
will  increase  rapidly  with  the  magnitude  of  the  angle 
measured,  and  the  adjustments  should  be  made  correct 
to  within  10'  if  the  resulting  errors  for  an  angle  of  90° 
are  to  be  insensible. 

However  carefully  these  adjustments  are  made  there 
will  remain  a  source  of  error  which  cannot  be  removed 
by  adjustment,  but  whose  effect  must  be  determined 
and  applied  as  a  correction  to  the  readings  if  the  maxi- 
mum attainable  precision  of  the  instrument  is  required. 
It  is  assumed  above  that  the  centre  of  the  graduated  arc 
falls  exactly  at  the  centre  of  motion  of  the  index -glass, 
but  the  maker  is  seldom  able  to  secure  this  exact  agree- 
ment, and  without  it  the  readings  of  the  vernier  are  not 
an  accurate  measure  of  the  amount  of  rotation  of  the 


INSTRUMENTS.  135 

mirror.  The  effect  of  this  error,  which  is  called  eccen- 
tricity, combined  with  the  effect  of  all  other  outstanding 
errors  of  the  instrument,  is  best  determined  by  carefully 
measuring  with  it  a  set  of  known  angles  of  different 
magnitudes,  from  o°  to  the  largest  one  possible,  and  treat- 
ing the  difference  between  the  measured  value  and  the 
true  value  of  each  angle  as  a  correction  to  the  corre- 
sponding reading  of  the  sextant.  These  corrections  may 
be  plotted  as  ordinates  with  the  sextant  readings  as 
abscissas  and  a  curve  drawn,  from  which  intermediate 
values  of  the  correction  may  be  read.  The  length  of 
the  arc  joining  two  stars  whose  right  ascensions  and 
declinations  are  given,  may  be  computed  and  used  as  a 
known  angle  for  this  purpose,  provided  the  effect  of 
refraction  in  altering  this  distance  is  duly  taken  into 
account;  or  if  a  distant  part  of  the  horizon  can  be  seen, 
a  set  of  angles  may  be  measured  with  a  good  theodolite 
for  comparison  with  the  sextant  results. 

58.  Index  Correction. — Since  the  value  of  the  index 
correction  for  very  distant  objects  is  constant  so  long 
as  the  adjustments  of  the  sextant  remain  unchanged, 
it  may  be  determined  from  special  observations  made 
for  this  purpose,  but  the  determination  should  be  fre- 
quently repeated  since  the  adjustment  is  easily  disturbed. 
Let  a  shade -glass  be  placed  over  the  eye  end  of  the  tele- 
scope and  the  direct  and  reflected  images  of  the  sun 
brought  into  contact,  externally  tangent  to  each  other, 
in  each  of  the  two  possible  positions,  H  first  right,  then 
left  of  /.  The  mean  of  the  corresponding  sextant  read- 
ings will  be  the  required  value  of  R0.  Since  the  index 


136  FIELD  ASTRONOMY. 

correction  enters  into  the  value  of  every  measured  angle, 
it  should  be  carefully  determined  from  several  settings, 
as  in  the  following  example: 

* 

DOUBLE   DIAMETER    OF   SUN   FOR   INDEX    CORRECTION. 

Caroline  Island,  April  22,  1883. 

Observer,  W.  U. 

On  Arc. 
o°  26'     o" 
26      o 

26          5 

26       o 

25      40 

25    40 


Off  Arc. 

Reduction. 

359°  22' 

5" 

#0=359°  53'  54" 

22 

0 

^  = 

+  6      6 

22 

o 

21 

40 

45  = 

i 

4      o 

21 

5° 

5  = 

o 

16       o 

21 

5° 

Almanac  = 

o 

15    56-4 

360°  25'   54"          359°  «'   54" 

In  place  of  subtracting  R0  from  each  subsequent 
reading  of  the  instrument  it  is  in  this  case  more  con- 
venient to  employ  the  quantity  2' =  360°  —  R0  as  a  correc- 
tion to  be  added.  The  readings  ' '  Off  Arc  "  were  taken  on 
the  supplementary  arc  to  the  right  of  the  o°,  and  the 
student  should  note  that  the  resulting  R0  falls  off  the  arc. 
Referring  to  the  position  of  RQ,  the  sign  of  the  index 
correction  may  be  determined  from  the  sailor's  rule: 
When  it  (RQ)  is  on  it's  off  (i  subtractive) ,  and  when  it  (R0) 
is  off  it's  on  (i  additive).  The  values  of  the  sun's  semi- 
diameter  furnished  by  the  observations  and  given  in  the 
almanac  are  shown  above  for  comparison. 

59.  Artificial  Horizon.  —  Altitudes  may  be  measured 
with  a  sextant  either  from  the  natural  (sea)  horizon  or 
from  an  artificial  horizon,  one  form  of  which  is  a  shallow 
box  containing  mercury,  covered  by  a  glass  roof  to  pro- 
tect it  from  wind.  The  reflecting  surface  of  the  liquid 
is  improved  by  adding  to  it  a  little  tin-foil  and  removing 


INSTRUMENTS.  137 

the  resulting  scum  (oxide)  with  the  edge  of  a  card. 
The  reflected  image  of  the  sun  or  star  lies  as  much  below 
the  true  horizon  as  the  real  object  is  above  it,  and  if 
the  angle  between  the  two  is  measured  with  the  sextant 
it  gives  at  once  the  double  altitude  of  the  body,  subject 
to  correction  for  index  error,  etc.  See  §  29  for  an  ex- 
ample. 

60.  Precepts  for  the  Use  of  a  Sextant. — i.  Keep   your 
fingers  off  the  graduation.     It  tarnishes  readily. 

2.  Focus  the  telescope  with  great  care  so  as  to  secure 
sharply  defined  images. 

3.  Make    the    direct    and    reflected    images  equally 
bright,  by  moving  the  telescope  to  or  from  the  plane  of 
the  sextant  with    the  adjusting-screw  provided  for  this 
purpose. 

4.  Bring  the  images  into  contact  midway  between 
the  guide -threads. 

5.  Don't  try  to  hold  the  images  still  in  the  field  of 
view.     Give   the   reflected   image   a   regular   oscillating 
motion  by  twisting  the  wrist,  and  note  its  relation  to  the 
direct  image  as  it  swings  by. 

6.  In  observing  the  sun  take  an  equal  number  of 
observations  on  each  limb  (edge). 

7.  Take  an  equal  number  of  observations  in  each 
position  of  the  horizon  roof,  direct  and  reversed. 

8.  Determine  the  index  correction  as  carefully  as  the 
angle  which  you  wish  to  measure. 

9.  Whenever  possible  use  a  shade-glass  over  the  eye- 
piece instead  of  those  attached  to  the  sextant  frame. 

10.  Work  as  rapidly  as  you  can  without  hurrying. 


138  FIELD  ASTRONOMY. 

6 1.  Chronometers. — This  section  will  be  confined  to 
a  consideration  of  the  proper  care  and  use  of  timepieces. 
For  an  account  of  their  mechanical  construction  see  the 
article  Watches  in  the  Encyclopaedia  Britannica. 

A  chronometer  is  a  large  and  finely  constructed 
watch,  whose  face,  hands,  and  train  (wheels)  are  to  be 
considered  as  a  mechanical  device  for  automatically 
counting  and  registering  the  vibrations  of  a  steel  helix, 
called  the  balance-spring.  In  most  chronometers  this 
spring  makes  one  complete  vibration  every  half  second, 
producing  a  beat  (tick)  of  the  chronometer  and  a  forward 
movement  of  the  seconds  hand  through  os.5.  This 
spring  may  vibrate  too  slow  or  too  fast,  thus  producing 
a  rate  of  the  chronometer,  and  it  is  practically  convenient 
that  this  rate  should  be  small,  but  the  real  test  of  excel- 
lence in  a  timepiece  is  not  the  magnitude  of  its  rate,  but 
its  uniformity  of  rate  from  day  to  day. 

In  order  that  the  rate  of  a  chronometer  shall  remain 
constant,  every  precaution  must  be  taken  against  dis- 
turbing the  balance-spring,  and  most  of  the  following 
precepts  for  the  treatment  of  a  chronometer  have  reference 
to  this  condition.  Of  the  various  mechanical  disturbances 
to  which  it  is  subject,  experience  shows  that  a  quick 
rotary  motion  about  the  axis  of  the  balance -spring  is  the 
most  injurious.  According  to  the  chronometer  makers 
a  single  quick  motion  of  this  kind  through  half  a  turn 
and  back  may  change  the  chronometer  correction  several 
seconds  and  so  disturb  the  rate  that  it  will  not  resume 
its  normal  value  for  hours  or  even  days. 

A  chronometer  is  usually  supported  in  gimbals  and 


INSTRUMENTS.  139 

should  be  allowed  to  swing  freely  in  them  when  at  rest, 
in  order  that  it  may  assume  a  vertical  position ;  but  when 
carried  about,  the  gimbals  should  be  locked  since  the 
oscillations  that  would  otherwise  be  imparted  to  the 
balance -spring  are  more  injurious  to  the  rate  than  the 
isolated  shocks  that  it  may  receive  when  firmly  held  in 
one  position.  A  chronometer  should  be  kept  in  a  dry 
place,  not  exposed  to  magnetic  influences.  If  possible 
it  should  always  rest  in  the  same  azimuth,  e.g.,  the  zero 
of  the  dial  always  pointing  north.  It  should  be  wound 
at  regular  intervals,  and  its  temperature  should  be  kept 
as  nearly  uniform  as  possible.  The  average  chronometer 
runs  best  at  a  temperature  near  70°  Fahr. 

62.  Comparison  of  Chronometers.  —  A  problem  of  fre- 
quent recurrence  is  the  comparison  of  one  chronometer 
with  another,  e.g.,  in  order  to  determine  the  correction 
of  one  from  the  known  value  of  AT  for  the  other.  This 
comparison  consists  in  noting  the  time  indicated  by  one 
chronometer  at  a  given  time  shown  by  the  other,  and 
presents  little  difficulty  when  no  greater  accuracy  than 
the  nearest  half -second  is  required.  If  the  comparison 
is  to  be  made  correct  to  the  nearest  os.i,  the  method 
of  coincident  beats  may  be  employed  if  one  of  the  chro- 
nometers keeps  sidereal  and  the  other  solar  time. 

Since  sidereal  time  gains  236  seconds  per  day  upon 
mean  solar  time  and  the  chronometers  beat  half -seconds, 
there  will  be  472  epochs  during  a  day,  at  which  the 
chronometers  beat  in  unison,  i.e.,  a  coincidence  of  the 
beats  occurs  every  three  minutes  throughout  the  day, 
and  if  the  comparison  be  made  at  one  of  these  coinci- 


HO  FIELD  ASTRONOMY. 

dences  by  noting  by  each  chronometer  its  indicated 
time  when  the  beats  are  coincident,  no  fractions  of  a 
second  need  be  determined  and  the  comparison  can  be 
made  correct  within  one  or  twqiiundredths  of  a  second. 

This  mode  of  comparison  is  illustrated  in  the  follow- 
ing example  of  the  comparison  of  two  mean -time  clocks, 
M  and  F,  with  each  other  by  comparing  each  with  a 
sidereal  clock  designated  H . 

OBSERVED    TIMES    OF    COINCIDENT    BEATS. 

M     i9h  3om  49s  F      igh  34™  55s 

H     10     42       i  H     10     46     o 

The  interval  between  the  coincidences,  as  measured  by 
H,  is  3m  59s  (sidereal),  and  this  interval  reduced  to  mean 
solar  units  and  added  to  M,  or  subtracted  from  F,  gives 
a  comparison  between  the  mean-time  clocks  as  follows : 

M     i9h  34m  473.35  i9h    30™  49*.oo 

F     19     34     55  .00  19     30     56  .65, 

either  form  showing  that  F  was  7s.  6  5  faster  than  M. 

Every  observer  should  acquire  the  ability  to  ' '  carry 
the  beat"  of  a  chronometer,  i.e.,  to  listen  to  and  count 
the  beats  while  attending  to  something  else,  since  nearly 
all  observations  in  which  it  is  required  to  note  the  time 
of  an  event,  e.g.,  the  transit  of  a  star  over  a  thread, 
require  this  ability  unless  special  mechanical  devices, 
such  as  a  chronograph,  are  employed.  (See  §  79.) 


CHAPTER  VIII. 
ACCURATE   DETERMINATIONS. 

63.  General  Principles.  —  Where  a  high  degree  of 
precision  is  desired  in  the  results  of  observation,  the 
purely  instrumental  sources  of  error  that  have  been 
examined  in  the  preceding  chapter  must  be  eliminated 
by  the  methods  there  shown,  or  by  others  equivalent 
to  them.  But  this  alone  is  not  sufficient,  and  we  note, 
for  example,  that  an  instrument  taken  from  a  .warm 
place  and  set  up  in  a  cold  one  undergoes  a  process  of 
cooling  and  contraction  of  its  parts  that,  while  in  prog- 
ress, renders  the  errors  of  adjustment  variable  quantities, 
whose  effects  cannot  be  represented  by  the  formulae 
derived  for  the  case  of  "instrumental  constants."  We 
have  therefore  as  a  rule  to  be  carefully  observed  when 
precision  is  required:  Let  the  instrument  be  set  up  and 
levelled  in  the  place  where  it  is  to  be  used,  at  least  half  an 
hour  before  observations  are  commenced.  Let  the  sur- 
roundings of  the  instrument  during*  this  period  be  as 
nearly  as  possible  like  those  under  which  the  observa- 
tions are  to  be  made,  i.e.,  shutters  open,  lamps  lighted, 
etc.  As  a  corollary  to  this  rule  we  have  the  further  pre- 
cept that  during  the  progress  of  the  observations  the 

141 


142  FIELD  ASTRONOMY. 

observer  and  his  lamp  should  be  kept  away  from  the 
instrument  as  much  as  possible. 

There  is  large  room  for  the  display  of  good  judgment 
in  the  selection  of  stars  to  be  observed  for  a  given  pur- 
pose, such  as  the  determination  of  time  or  azimuth,  and 
precepts  bearing  upon  this  choice,  both  with  reference 
to  the  precision  of  the  observations  themselves  and  to 
the  elimination  of  errors  in  the  right  ascensions  and 
declinations  of  the  stars  as  furnished  by  the  almanac,  are 
given  in  the  following  sections. 

Whenever  observations  are  to  be  made  upon  a  con- 
siderable number  of  different  stars,  as  in  determinations 
of  time  and  latitude,  an  observing  list  should  be  prepared 
in  advance,  giving  the  names  and  magnitudes  of  the  stars, 
arranged  in  the  order  in  which  they  are  to  be  observed, 
and  giving  also  such  data  as  may  be  required  for  finding 
them  with  the  given  instrument,  e.g.,  their  right  ascen- 
sions, declinations,  zenith  distances,  etc.  Also,  a  form 
should  be  prepared  in  which  to  record  the  observations, 
each  figure  that  is  to  be  written  down  as  a  part  of  the 
record  having  its  proper  place  allotted  it.  This  place 
must  be  filled  up  before  the  observation  is  complete, 
and  the  presence  of  an  unfilled  space  in  the  form  is  to 
be  considered  as  a  reminder  that  something  remains  to 
be  done. 

64.  Time  by  Equal  Altitudes.  —  The  best  method  of 
determining  time  involves  the  use  of  a  transit  instru- 
ment (see  Chapter  IX),  but  an  excellent  time  determina- 
tion may  be  made  with  a  theodolite,  zenith  telescope, 
or  sextant  by  the  method  of  equal  altitudes,  as  follows : 


ACCURATE  DETERMINATIONS.  143 

We  note  the  chronometer  time,  7\,  at  which  a  star 
west  of  the  meridian  reaches  the  zenith  distance  zl  and 
the  time,  T2,  at  which  another  star,  east  of  the  meridian, 
reaches  a  zenith  distance,  02,  which  differs  as  little  as 
possible  from  zr  In  sextant  observing  it  is  customary 
to  assume  that  if  the  sextant  is  set  to  the  same  reading 
in  the  two  observations  we  shall  have  z1=z2.  For  an 
instrument  of  the  other  type  (theodolite)  the  telescope 
must  be  left  firmly  clamped  in  altitude  as  it  is  turned 
from  one  object  to  the  other,  and  any  slight  change  in 
the  altitude  of  the  line  of  sight  must  be  carefully  deter- 
mined from  readings  of  the  altitude  level  of  the  instru- 
ment. If  the  bubble  changes  its  position  in  the  level- 
tube  when  the  latter  is  turned  from  the  first  to  the  second 
star,  it  should  be  brought  back  to  its  original  place  by 
the  levelling  screws  of  the  instrument,  but  the  angle 
between  the  telescope  and  level-tube  must  not  be  altered. 
If  the  instrument  is  provided  with  an  azimuth  circle,  it 
will  be  well  to  note  its  readings,  R^  and  R2  ,  correspond- 
ing to  the  observed  Tl  and  T2. 

For  the  reduction  of  the  observations  we  take  from 
the  formulae  for  transformation  of  coordinates,  §  14,  the 
equations 

cos  zt  =  sin  0  sin  dl  +  cos  0  cos  <^  cos  ^  , 
cos  z2  =  sin  0  sin  d2  +  cos  0  cos  d2  cos  t2  , 

and  in  these  relations  if  we  could  assume, 


we  should  have  at  once, 

cos  J,  =  cos  t2    and     ^  =  —  12. 


144  FIELD  ASTRONOMY. 

From  this  last  relation  we  obtain, 


and  solving  this  for  J!T,  find, 

jr  =  i(ai  +  a2)-i(ri  +  T2).  (104) 

This  ideal  case  may  be  realized  in  practice  by  observing 
the  times  at  which  a  given  star  comes  to  equal  zenith 
distances  on  opposite  sides  of  the  meridian,  i.e.,  before 
and  after  its  culmination,  but  this  may  involve  a  delay 
of  several  hours  between  the  observations,  and  it  will 
usually  be  more  convenient  and  expeditious  to  observe 
in  quick  succession  two  stars  of  nearly  equal  declination 
but  widely  different  right  ascension,  one  east  and  the 
other  west  of  the  meridian,  or  the  sun,  A.M.  and  P.M. 

To  adapt  Equation  104  to  this  case  we  assume  six 
new  quantities,  z,  B,  d,  D,  t,  and  L,  defined  by  the  fol- 
lowing relations: 


z-B=z2,     3-D  =  d2,     t-L  =  t2, 

and  from  the  last  pair  of  these  equations  we  obtain,  by 
the  method  followed  in  deriving  Equation  104, 

AT  =  JK  +  «2)  -  i(7\  +  T2)  +  L.  (106) 

To  determine  the  value  of  L  we  introduce  into  Equations 
103  the  quantities  defined  in  Equations  105,  and  sub- 
tracting the  first  of  these  transformed   equations   from 
the  second,  obtain  the  rigorous  relation, 
sin  z  sin  B  =  —  sin  0  cos  d  sin  D 

+  cos  0  sin  d  sin  D  cos  t  cos  L 

+  cos  0  cos  d  cos  D  sin  /  sin  L.      (107) 


ACCURATE  DETERMINATIONS.  145 

This  equation  is  quite  too  cumbrous  for  use,  but  if  in  the 
plan  and  execution  of  the  observations  care  is  taken  to 
make  B  and  D  small  quantities  whose  cubes  and  higher 
powers  may  be  neglected,  it  is  readily  reduced  to  the 
simpler  form, 

B 


sm  t          tan  t          cos  0  sin  A 
From  Equations  105  we  find  for  use  here, 


It  appears  from  these  relations  that  the  quantity  B  is 
half  the  change  of  zenith  distance  suffered  by  the  line 
of  sight  in  passing  from  one  star  to  the  other,  and  this 
change  should  be  measured  with  all  possible  care  by 
means  of  the  altitude  level  of  the  instrument.  If  we 
represent  by  b  the  observed  displacement  of  the  bubble 
between  the  two  observations  and  by  d  the  value  of  half 
a  level  division,  we  shall  have 

B=±bd,  (no) 

where  the  positive  sign  is  to  be  used  when  the  bubble 
stands  nearer  to  the  objective  end  of  the  telescope  at 
the  eastern  than  at  the  western  observation.  The 
value  of  B  is  required  in  seconds  of  time,  and  it  will  there- 
fore be  convenient  to  express  d  in  terms  of  the  same  unit 
instead  of  in  seconds  of  arc. 

The  declination  factor,  D,  should  also  be  expressed 
in  seconds  of  time,  and  since  declinations  are  usually 
given  in  arc,  we  reduce  the  difference  dl  —  d2  to  seconds 


146 


FIELD  ASTRONOMY. 


of  -arc,  and  dividing  this  by  1 5  obtain  in  terms  of  the 
required  unit 

tf-Atfi-A)"-  (in) 

65.  Example. — Time  by  Equal  Altitudes. — The  follow- 
ing example  illustrates  the  application  of  these  equations 
to  the  reduction  of  observations  made  with  an  engineer's 
transit  provided  with  stadia  threads,  over  which  the 
star's  vertical  transits  were  observed,  the  instrument 
being  turned  between  times  so  that  the  transit  over  the 
horizontal  thread  should  always  occur  near  its  inter- 
section with  the  vertical  thread.  The  three  terms  con- 
tained in  the  value  of  L  (Equation  108)  are  here  repre- 
sented by  the  symbols  Lv  L2,  L3. 

EQUAL   ALTITUDES    FOR   TIME. 

Thursday,  April  30,  1896. 
At  Brick  Pier.     Instrument,  Heyde.     Observer,  C. 


Star 

a  Orionis 

a  Serpentis 

Obs'd  T 

ioh  41'"    9s.  6 

ioh  50™  42*.  i 

*  a\  +  7"i) 

ioh45m  55s-  8 

4i      37-2 

50      14.2 

55-7 

42        5.2 

49      45-3 

55-2 

Level* 

19-7      3-3 

18.8      2.6 

—  Mean 

-io    45    55-57 

R 
d 

83°  Si' 
+  7   23   18.8 

277°  23' 
4-6  44   51.3 

L 
i(«i  +  «2) 

+  1      12.  II 

+  10    44    22.24 

a 

5h  49m  33s-°6 

i5h  39m  ii8-  42 

AT 

—  0     21.22 

*  The  end  of  the  bubble  nearer  the  objective  is  recorded  first,     d  =  3".8  =  o'.25. 


K«2—  ai) 

4h  54m  49s 

•ir-i 

4-0°  38'  27" 

5     b\—  *>* 

-o.Srf 

i(7"2  —  7\) 

—  4      18 

D 

+-76S.92 

B 

9  .  3oiow 

t  (time) 

4   5°     31 

tan  <£ 

9.9709 

sec  4> 

0.1364 

*  (arc) 
a 

72°  38' 
7       4 

cosec  * 
£> 

0.0203 
1.8860 

cosec  A 
logL3 

0.0032 
9  .4406^ 

43       5 

cot  t 

9-4952 

•l(Ri-RJ 

83     14 

—tan  5 
logL, 

9-°933w 
1.8772 

L1 

+  75-37 
—   2   98 

logL, 

0-4745" 

L, 

—   0.28 

ACCURATE  DETERMINATIONS.  147 

For  the  sake  of  illustration  the  reductions  in  the  pre- 
ceding example  are  carried  to  hundredths  of  a  second  of 
time,  but  this  is  a  quantity  quite  inappreciable  in  the 
telescope  of  an  engineer's  transit,  and  with  such  an  instru- 
ment, or  with  a  sextant,  it  will  usually  be  sufficient  to 
carry  the  reductions  to  tenths  of  seconds  only.  Corre- 
sponding to  this  degree  of  accuracy  the  difference  of 
declination  of  the  stars  may  be  as  great  as  two  or  three 
degrees  without  the  introduction  of  sensible  error  into 
the  results  by  reason  of  the  approximate  character  of 
the  reduction  formulas.  The  difference  should  not  ex- 
ceed one  half  of  this  amount  if  hundredths  of  seconds 
are  to  be  taken  into  account. 

66.  Observing  List. — Without  transgressing  these  rather  narrow 
limits  for  dl  —  $2,  a  considerable  number  of  suitable  pairs  of  stars  may 
be  selected  from  the  almanac,  as  is  illustrated  by  the  short  observing 
list  given  below,  and  such  a  list  should  be  prepared  for  the  particular 
time  and  place  at  which  observations  are  to  be  made.  At  least  one 
of  the  stars  in  each  pair  should  be  a  bright  one,  easily  recognized  and 
found  with  the  telescope  by  sighting  over  its  tube.  The  second  star 
of  the  pair,  even  though  much  fainter,  may  be  readily  found  by  the 
method  given  below. 

In  the  selection  of  pairs  of  stars  care  should  be  taken  to  secure  those 
that  are  as  near  as  may  be  to  the  prime  vertical  at  the  time  when 
their  altitudes  are  equal,  since  the  motion  in  altitude  is  then  most 
rapid  and  most  accurately  observed.  The  analytical  expression  for 
this  condition  is 

tan  £ (d,  +  <?2)  =  tan  $  cos  £(«i  - «2)  i  C1  x 2) 

and  if  this  equation  is  satisfied  by  the  coordinates  of  any  two  stars 
that  differ  but  little  in  declination,  these  stars  will  be  near  the  prime 
vertical  at  the  instant  when  their  altitudes  are  equal.  But  this  con- 
dition should  not  be  too  rigorously  insisted  upon,  and  even  considerable 
deviations  from  it  may  be  permitted  in  order  to  secure  a  suitable 
number  of  bright  stars. 

Having  chosen  a  pair  of  stars,  we  may  determine  as  follows  the 
sidereal  time,  6,  at  which  their  altitudes  will  be  equal:  In  Equation  106 
we  put  AT  =o,  T1  =T2  =0,  and  obtain 

£»  ("3) 


148  FIELD  ASTRONOMY. 

where  the  value  of  L  is  to  be  derived  from  Equation  108,  omitting 
the  term  in  B.  It  will  usually  be  convenient  to  observe  the  first  star 
about  five  minutes  before  the  computed  time,  '. 

Finding  the  Faint  Star.  —  If  the  two  stars  have  equal  declinations, 
their  azimuths  at  the  instant  of  equal  altitudes  will  be  numerically 
equal  but  of  opposite  sign,  i.e.,  A1-^A2=o,  while  if  their  declinations, 
differ  slightly,  there  will  be  a  small  difference  in  the  azimuths  which 
will  transform  this  equation  into 

At+A9+dA3*=o.  (114) 

To  determine  the  value  of  dA2  in  this  equation  we  obtain  from  the 
astronomical  triangle  the  relation  (Equation  15) 

sin  d  =cos  z  sin  <f>—  sin  z  cos  9*  cos  A,  (IJ5) 

and  differentiating  this,  treating  9"  and  z  as  constants,  we  find  as  the 
change  of  azimuth  of  the  second  star  produced  by  a  small  change  of 
declination, 

cos  d  dd  =cos  9"  sin  z  sin  A  dA  =cos  9"  cos  d  sin  t  dA  ,         (i  16) 
from  which, 

dA  =       d*  .    ,.  (117) 

cos  9  sm  / 

Let  RI,  R2,  RQ  represent  respectively  the  readings  of  the  horizontal 
circle  when  the  telescope  is  directed  to  the  western  star,  to  the  eastern 
star,  and  to  the  meridian;  we  shall  then  have 

A^Rt-Ro,         A2=R2-R0,  (118) 

and  substituting  in  Equation  114  these  relations  together  with  the 
approximate  values, 

dd  =  d2-dlf         *=i(aa-ai),  (119) 

we  obtain 

r.  (120) 


l2  Q       -  -r-.  -  r. 

cos  9  sin  £(a2  —  «i) 

The  last  term  in  this  expression,  computed  for  9*  =43°,  is  tabulated 
in  the  observing  list  under  the  heading  AR,  and  by  means  of  it  and 
the  reading  Rl  to  the  first  star  of  a  pair,  the  reading  of  the  horizontal 
circle,  R2,  may  be  found  at  which  the  instrument  should  be  set  and  the 
arrival  of  the  second  star  in  the  field  awaited  at  a  time  as  much  after 
the  computed  6  as  the  first  observed  time  was  earlier  than  6.  For 
convenience'  sake  orient  the  instrument  and  make  R0  =  o.  As  a 
control  upon  the  sign  of  AR,  note  that  the  star  that  has  the  larger 
declination  must  be  the  farther  from  the  south  point. 


ACCURATE  DETERMINATIONS. 


149 


TIME    BY    EQUAL    ALTITUDES. 
Partial  Observing  List  for  $  =43°. 


Stars. 

Mag, 

R.  A. 

Dec. 

0. 

AR. 

o,  Orionis 

o  o 

h.       m. 
c       tro 

0                    / 

7        23 

h.     m. 

0                / 

a  Serpentis 

2     7 

I  IT          70 

6      4<? 

10     44 

-o      54 

a  Can  Min  

o    s 

7       34 

C       2Q 

a  Serpentis  
ft  Geminonim  

2-3 
I     2 

15     39 
7       30 

6     45 
28      16 

H      34 

+  2           0 

fi  Herculis  

7     e 

17      4? 

27      47 

12      43 

-o     41 

a  Leonis  

I  .  7 

IO         3 

12        28 

a  Ophiuchi  

2  .  2 

17       3O 

12        38 

13      46 

+  o      16 

o,  AouilcC 

O    O 

1  0       4.6 

8      3.6 

o  Leonis 

3  8 

936 

IO        21 

14     44 

-2       28 

67.  Precise  Azimuths.  —  The  azimuth  of  a  terrestrial 
line,  e.g.,  the  line  joining  the  centre  of  a  theodolite  to  a 
distant  mark,  may  be  determined  by  measuring  the 
difference  of  azimuth,  D,  between  the  mark  and  a  star 
at  an  observed  time,  T.  From  the  observed  time  and 
the  right  ascension  of  the  star  its  hour  angle,  /,  may  be 
derived,  and  from  Equations  14  we  obtain,  by  division 
and  introduction  of  the  auxiliary  quantities, 


the  relation 


-  tan  A  = 


gsin/ 


(121) 


(122) 


I—  k  COS  t1 

from  which  the  true  azimuth  of  the  star  at  the  time  of 
observation  is  readily  computed.     The  azimuth  of  the 

mark  is  then 

A'=A+D,  (123) 

where  D  is  assumed  to  be  measured  from  the  star  toward 
the  east. 


150  FIELD  ASTRONOMY. 

The  precision  of  A'  depends  equally  upon  D  and  A, 
and  through  A  it  depends  upon  the  assumed  latitude, 
declination,  right  ascension,  and  chronometer  correction 
that  are  employed  in  the  computation.  The  observa- 
tions should  therefore  be  planned  with  reference  to  elim- 
inating whatever  minute  error  may  exist  in  any  of  these 
data,  and  to  overcoming,  by  the  methods  indicated 
below  and  in  §  54,  the  effect  of  instrumental  errors  upon 
the  measured  angle  D. 

Errors  in  the  Assumed  Data. — The  effect  of  these  errors 
may  be  greatly  diminished  by  selecting  for  observation 
a  star  very  near  the  pole  of  the  heavens,  since  the  factor 
g  is  thus  made  small,  and  such  a  star,  e.g.  Polaris,  should 
always  be  chosen.  If  the  chronometer  correction  is  well 
determined,  the  observations  may  be  made  at  any  con- 
venient hour,  whether  near  elongation  or  not.  As  a  guide 
to  the  required  precision  in  AT  we  note  that  for  observa- 
tions of  Polaris  within  the  limits  of  the  United  States 
an  error  of  2s  in  the  time  will  in  no  case  produce  in  the 
computed  azimuth  an  error  greater  than  i". 

If  the  highest  precision  is  required,  the  star  should  be 
observed  at  two  points  of  its  diurnal  path  which  are 
diametrically  opposite  to  each  other,  i.e.,  there  should 
be  two  groups  of  observations  separated  by  an  interval 
of  twelve  hours,  or  some  odd  multiple  of  twelve  hours. 
Errors  in  0,  d,  and  a  will  then  be  almost  perfectly  elim- 
inated, and  there  will  also  be  eliminated  any  systematic 
personal  error  of  observation  depending  upon  the  direc- 
tion of  the  star's  apparent  motion,  such  as  is  sometimes 
found  to  exist  in  the  work  of  even  the  best  observers. 


ACCURATE  DETERMINATIONS.  151 

A  similar  but  less  complete  elimination  of  errors  may  be 
obtained  from  observations  made  at  a  single  epoch  if 
these  are  equally  divided  between  stars  upon  opposite 
sides  of  the  pole  and  equidistant  from  it.  Examples  of 
pairs  of  stars  which  approximately  fulfil  this  condition 
are  Polaris  and  6  Ursse  Minoris;  51  H.  Cephei  and 
d  Ursse  Minoris. 

The  angle  D  may  be  measured  with  either  a  repeating 
or  a  non-repeating  (direction)  instrument,  and  the  student 
should  observe  the  following  respects  in  which  their  use 
differs:  For  a  repeating  instrument  the  azimuth  level 
should  be  used  to  determine  the  inclination  of  the  verti- 
cal axis  corresponding  to  the  lower  motion  of  the  instru- 
ment. For  a  non-repeating  instrument  the  inclination 
to  be  determined  is  tha.t  of  the  horizontal  axis.  In  both 
cases  the  bubble  readings  are  to  be  taken  when  the  line 
of  sight  is  directed  toward  the  star  and  also  when  it  is 
turned  toward  the  mark,  unless  the  latter  has  a  zenith 
distance  of  90°,  in  which  case  erroneous  levelling  will  not 
affect  the  readings  to  it. 

With  any  type  of  instrument  the  horizontal  circle 
is  to  be  turned  in  its  own  plane  from  time  to  time  during 
the  observations,  so  that  the  vernier  or  microscope  read- 
ings shall  be  symmetrically  distributed  throughout  the 
entire  360°  of  the  graduation ;  e.g.,  for  an  instrument  with 
two  microscopes  let  one  ninth  of  the  total  number  of 
observations  be  made  with  the  circle  reading  to  the 
mark  approximately  o°,  another  ninth  with  the  circle 
reading  20°,  46°,  60°,  etc.  But  see  §  53  for  the  peculiar 


152  FIELD  ASTRONOMY. 

manner  in  which  the  circle  settings  should  be  changed 
in  the  case  of  a  repeating  instrument. 

Level  Corrections.-^The  correction  for  level  error  is 
to  be  applied  to  each  circle  'reading  as  shown  in  §  50, 
but  for  observations  made  by  the  method  of  repetitions 
the  level  correction,  b'  tan  h,  there  given  for  the  reading 
to  the  star,  must  be  multiplied  by  n,  the  number  of  point- 
ings in  a  set,  since  the  difference  of  the  corrected  read- 
ings to  star  and  mark  is  to  be  divided  by  n  in  order  to 
obtain  the  measured  angle.  It  will  usually  be  expedient 
to  arrange  the  form  of  record  of  the  observations  so  that 
the  level  corrections  may  be  applied  and  the  angles  worked 
out  in  the  record  book. 

68.  Reduction  of  the  Observations.  —  After  the  hour 
angles  have  been  formed  from  the  relation  t  =  T  +  4T—a, 
and  the  constants  g  and  k  computed  with  the  known  decli- 
nation and  latitude,  the  computation  of  A  presents  no 
difficulties,  but  it  may  be  considerably  abbreviated 
through  the  use  of  Albrecht's  Tables  (reproduced  in 
Appendix  VII,  Annual  Report  U.  S.  Coast  and  Geodetic 
Survey,  1897-98),  which  with  the  argument  log  x  give 

the  logarithm  of     ^    .     Calling  this  last  factor  F,  and 

putting  k  cos  t=x,  Equation  122  assumes  the  very 
simple  form 

—  tan  A  =  gF  sin  t.  (124) 

In  the  absence  of  special  tables  for  F  its  value  may 
be  readily  obtained  from  an  ordinary  table  of  addition 
and  subtraction  logarithms  as  follows:  Representing 


ACCURATE  DETERMINATIONS.  153 

by  A  and  B  the  argument  and  function  in  such  a  table,  * 
i.e.,  A=  log  x,  £=log(i+tf),  we  have,  whenever  cos  t 
is  negative,  A  =log  (k  cos  t),  log  F  =  —  B.  When  cos  / 
is  positive,  we  use  the  development 


and  interpolating  from  the  table  of  addition  logarithms 
the  values  of  Blt  B2,  Bv  etc.,  corresponding  to  the  argu- 
ments x,  x2,  x4,  we  find 


log  F  =  J^  +  Ba  +  ^H-  etc.  (125) 

For  observations  of  Polaris  made  within  the  limits  of  the  United 
States  it  will  never  be  necessary  to  use  more  than  the  first  two  terms 
of  this  series,  e.g.,  corresponding  to  this  case  the  greatest  possible 
value  of  k  cos  t  furnishes  log  x  and  the  several  values  of  B  given  below: 

log*     8.39386     5t  0.0106248 
log*2    6.7877       B2  2663 

log*4    3.575         B,  _  2^ 

logF  0.0108913 

In  ordinary   practice  the  value  of  log  F  will  be  required  to  only  six 
places  of  decimals,  and  Bt+B2  furnishes  this  degree  of  precision, 

Where  the  highest  degree  of  precision  is  sought,  it  is 
customary  in  the  reduction  of  the  observations  to  com- 
pute for  each  observed  time  the  corresponding  value  of 
A,  but  this  process  may  be  very  greatly  abridged  by 
treating  the  mean  of  a  considerable  number  of  observa- 
tions as  a  single  observation  made  at  T0,  the  mean  of  the 
recorded  times.  The  azimuth  A0  computed  from  T0 
will  not  correspond  exactly  to  the  observations,  but  the 
correction  required  on  this  account  is  readily  obtained. 

*  Do  not  confound  this  use  of  A  with  its  wholly  different  meaning 
in  Equation  124. 


154  FIELD  ASTRONOMY. 

We  may  develop  by  Taylor's  Formula  the  relation  be* 
tween  azimuth  and  time  in  the  form, 


an  equation  which  obtains  for  each  observed  T  and  its 
corresponding  A.  If  we  take  the  mean  of  these  several 
equations  and  note  that  the  mean  of  the  (T—  T0)s  is 
necessarily  zero,  since  T0  is  the  mean  of  the  Ts,  we  find 
for  the  average  of  the  set, 


(126) 

where  the  last  term  of  the  expression  is  the  required 
correction  to  reduce  AQ  to  the  mean  of  the  observed 
azimuths.  For  the  numerical  application  of  this  formula 
we  need  to  introduce  a  convenient  expression  for  /"(A0), 
and  there  must  also  be  a  numerical  factor  such  that  the 
value  of  the  term  shall  be  given  in  seconds  of  arc  when 
T—T0  is  expressed  in  minutes  of  time.  This  factor, 
combined  with  the  coefficient  -J  which  appears  in  the 
equation,  is  readily  shown  to  be, 

6oXis\2     206265 

x—  —  =  [o-293°]- 

The  differential  coefficient,  f"(AQ),  does  not  admit 
of  an  expression  that  is  both  simple  and  rigorous,*  but, 
with  entire  accuracy  at  the  pole  and  approximately  for 
any  star  near  the  pole,  we  may  write 


*  The  complete  expression  for  f"(A0)  is 

d*A 

dt* 


—cos2  <£  sin  A  { (sec2  h  +  tan2  ti)cos  A  +  tan  <£  tan.fr}, 


ACCURATE  DETERMINATIONS.  155 

and  combining  these  several  expressions  we  find  as  the 
.correction  to  the  computed  azimuth,  AQ, 


4A0  =  +[0.2930]  sin  AZ(T-TQY,          (127) 

where  n  is  the  number  of  Ts  included  in  the  mean,  T0, 
and  the  differences,  T  —  T0,  are  to  be  expressed  in  minutes 
of  time.  4A0  must  always  be  so  applied  as  to  bring 
the  computed  A  0  nearer  to  the  meridian. 

See    §  85    for   the  extremely  small  effect  of  diurnal 
aberration  upon  azimuth  determinations. 

69.  Example.  —  Precise  Azimuth.  —  The  example   on 
p.  156  represents  a  determination  of  azimuth  made  with 
an  engineer's  transit,  using  the  method  of  repetitions, 
four  pointings  in  a  set,  and  combining  two  sets  in  such 
a  way  as  to  eliminate  the  effect  of  lack  of  parallelism  of 
the  axes  of  the  instrument,  see  §  53.     The  graduation 
errors  are  not  here  eliminated,  and  other  sets  with  read- 
ings symmetrically  distributed  about  the  circle  are  re- 
quired for  this  purpose. 

70.  Precise  Latitudes.  —  Zenith  Telescope  Method.  —  In 
Fig.  12  let    V  represent  any  point  on  the  meridian,  St 


FIG.  12.— Zenith  Telescope  Latitudes. 

and  52the  points,  on  opposite  sides  of  V,  at  which  two 
stars,  of  declination  ^  and  d2  respectively,  cross  ther 
meridian  in  their  diurnal  motion,  and  let  zl  and  z9  denote!. 


156 


FIELD  ASTRONOMY. 


PRECISE   AZIMUTH    DETERMINATION. 

At  Station  M.     Monday,  May  i,  1899. 
Instrument  No.  386.     Chronometer,  S.     Observer,  C. 
Chronometer  AT  =  —21n^gaij.     4  tan  h.d  =  io",j. 


Object. 

Circle 
«. 

L 
L 

4 
R 

R 
4 

Chronometer. 

Horizontal  Circle. 

Vert.  Circle 
Levels. 

Angle. 

Ver.  A. 

Ver.  B. 

Mark  
Polaris.  . 

Polaris  .  . 
Mark.  .  .  . 

h.     m.     s. 

12     25 
12     27     28 

30  59 
33   28 
36   24 

O           f         /t 

150    31    25 

117    21    4O 
117    22       0 

150  10     o 

/      // 

31    15 

21  35 

21     30 

9  40 

0            / 

0       0 
W.        E. 
8.5    15.0 
12.7   ii  .0 

o         tit 
150    31     20 

=               -26 
117     21     38 

-2.4 

4i     5° 

4i      43 
6.3      17.4 
18.0        5.8 

)l28     IQ 

)33    10     8 

12     32        4.8 

12  40     7 
45   28 
47   38 
49  40 

8   17   32 
117   21   45 

+  6 
150     9   50 

+  Q-55 

0       0 

12    52    .. 

)i82   53 

)32   47   59 

12  45  43.2 

8    12       0 

REDUCTION. 

h.    m.      s. 

h.    m.     s. 

t 

a 

43     4  37 
88  46   12.5 

I     21     28.O 

T  +  JT 
T  +  AT-a 
t 

12    29   25.1 

11      7   57  -1 
1  66°  5  9'  1  6" 

12   43      3-5 
11    21    35.5 

I70023'52" 

sec  <£ 
cot  d 

0.13641 
8.33180 

cos  t 

k  COS  t 

9.98870™ 
8.29132™ 

9  .99388™ 
8  .  29650™ 

tan  <£ 

9.97082 
8.46821 

sin  t 
g 

9-35249 
8.46821 

9  .  22221 
8.46821 

k 

8.30262 

F-.-B, 

9-99*59 

9.99149 

—tan  A0 

7.81229 

7.68191 

(T  —  T  o) 

2O,  I,  2,  l8 

;o,  o,  4,  16 

AQ 

179  37  4i 

179  43   28 

[2(T  —  TO) 

1  .009 

1.097 

AAQ 

o 

0 

sin  A0 

7.812™ 

7.682™ 

Const. 

0.293™ 

0.293™ 

D 

8  17  32 

8    12       0 

log  J^o 

9.114 

9.072 

4A0 

+  0".I3 

+  0".I2 

A' 

187  55   i3 

187  55  28 

The  corrections  JA0  computed  above  are  too  small  to  be  taken 
into  account  in  observations  of  this  character,  but  with  a  larger  instru- 
ment or  when  the  star  is  near  elongation  they  become  of  sensible 
magnitude. 


ACCURATE  DETERMINATIONS.  157 

the  arcs  VS^  and  VS2.     Denoting  by  <t>"  the  declination 
of  V,  we  have  from  the  figure 


and  by  subtraction, 

2<P'  =  (dl  +  d9)  +  (zl-za).  (128) 

Since  V,  by  supposition,  is  any  point  of  the  meridian, 
we  may  now  define  it  as  the  projection  upon  the  merid- 
ian, of  the  point  in  which  the  vertical  axis  of  a  theod- 
olite or  other  similar  instrument  meets  the  celestial 
sphere,  and  we  may  represent  by  6"  the  zenith  distance 
of  V,  reckoned  positive  when  the  zenith  lies  between  V 
and  the  pole.  Since  the  latitude  is  equal  to  the  declina- 
tion of  the  zenith,  we  shall  have 

20  =  2(0"  +  &'0  =  (tf1  +  aa+26/0  +  (*1-*3).  (129) 
In  the  practice  of  American  government  surveys 
all  precise  determinations  of  latitude  are  based  upon 
this  equation  and  are  made  with  an  instrument,  the 
zenith  telescope,  especially  designed  for  the  micrometric 
measurement  of  small  differences  of  zenith  distance, 
the  z1  —  z2  of  the  equation.  But  Equation  129  may  be 
applied  with  any  instrument  capable  of  measuring  alti- 
tudes— theodolite,  sextant,  etc. — and  in  general  it  will 
furnish  better  results  than  other  modes  of  using  the 
instrument,  since  if  the  stars  are  so  selected  that  zl 
differs  but  little  from  z2,  any  constant  errors  which  may 
be  present  in  the  instrumental  work  will  be  very  nearly 
the  same  for  the  two  stars,  and  will  be  approximately 
eliminated  from  the  difference  zl  —  z2.  We  shall  here 


158  FIELD  ASTRONOMY. 


develop  the  zenith-telescope  method  with  reference  to 
its  use  with  an  engineer's  transit  provided  with  a  gradi- 
enter  and  an  altitude  level*  which  latter  may  be  its 
striding-level  properly  fastened  to  the  alidade  at  right 
angles  to  the  horizontal  axis.  With  very  small  modi- 
fications the  resulting  formulae  will  be  applicable  to  the 
zenith  telescope  as  usually  constructed. 

The  first  step  in  the  application  of  the  method  is  the 
selection  of  an  observing  programme,  consisting  of  a 
number  of  pairs  of  stars  whose  right  ascensions  and 
declinations,  for  each  pair,  satisfy  the  conditions 


a2-a1<2om,        da  +  dl-2<f><±G,  (130) 

where  G  denotes  the  greatest  angle  that  can  be  con- 
veniently measured  with  the  gradienter.  Write  upon 
the  edge  of  a  slip  of  paper  the  approximate  value  of  2  0, 
and  turning  to  a  suitable  list  of  stars,  e.g.,  the  list  of 
mean  places  given  in  the  almanac,  subtract  each  decli- 
nation in  turn  from  20  and  seek  within  the  given  limits 
of  right  ascension  a  star  whose  declination  differs  but 
little  from  the  difference  thus  obtained.  If  bright  enough 
to  be  observed  with  the  given  instrument,  any  two  stars 
thus  related  will  constitute  a  latitude  pair. 

Having  prepared  such  an  observing  list,  before  the 
first  of  these  stars  comes  to  the  meridian  let  the  instru- 
ment be  carefully  levelled  and  oriented  and  its  telescope 
set  to  the  approximate  zenith  distance  of  the  star, 
zl  =  ±(4>—d1),  .  When  the  star  by  its  diurnal  motion 
is  brought  into  the  field  and  passes  behind  the  vertical 
thread,  a  pointing  in  altitude  should  be  made  upon  it 


PLATE  V. 


A  Zenith  Telescope  as  used  at  the  International  Latitude  Stations.     Length  <>f 
Telescope  52  inches.     Approximate  Cost  $1600. 

[To  face  p.  158.] 


ACCURATE  DETERMINATIONS,  159 

with  the  gradienter,  and  the  readings  of  the  altitude 
level  and  gradienter  head  recorded  immediately  after 
the  pointing.  Leaving  the  telescope  firmly  clamped 
in  altitude,  let  it  be  now  revolved  180°  in  azimuth  with- 
out loosing  the  altitude  clamp,  and  with  the  gradienter 
bring  the  line  of  sight  to  the  zenith  distance  of  the  second 
star,  z2=:F(0— <y>  and  observe  it  precisely  as  before. 
If  the  level-bubble  changes  its  position  in  the  tube  as  the 
instrument  is  turned  from  the  first  to  the  second  star,  it 
should  be  brought  back  to  its  initial  position  by  means 
of  the  levelling  screws. 

The  readings  of  the  level  in  the  two  positions  deter- 
mine the  average  value  of  6",  and  if  Rl  and  R2  represent 
the  respective  gradienter  readings  and  k  is  the  angle 
moved  over  by  the  line  of  sight  when  the  gradienter  is 
turned  through  one  complete  revolution,  we  shall  have, 

z1-z2=±k(Rl-R2).  (131) 

71.  Minor  Corrections. — Before  introducing  this  value 
into  the  expression  for  20  we  proceed  to  examine  some 
matters  that  require  further  explanation,  viz. : 

Level  Error. — The  small  term  26"  arises  from  a  devia- 
tion of  the  vertical  axis  of  the  instrument  from  the  true 
vertical  Its  amount  and  sign  are  to  be  determined 
from  readings  of  an  altitude  level,  as  shown  in  §  42. 
Make  this  error  small  by  turning  the  levelling  screws, 
if  necessary,  so  that  the  bubble  readings  shall  be  the  same 
for  the  second  star  as  for  the  first. 

Refraction. — The  effect  of  refraction  upon  the  latitude 
observations  is  most  readily  determined  by  substituting, 


160 


FIELD  ASTRONOMY. 


in  place  of  the  true  decimations  of  the  stars,  their  appar- 
ent declinations  as  affected  by  the  refraction.     This  dis- 
places each  star  toward  the  zenith  by  the  amount,  (§23) 
'   982" 


and  since  for  the  southern  star  this  displacement  in- 
creases, while  for  the  northern  star  it  diminishes,  the 
declination,  we  shall  have  as  the  sum  of  the  apparent 
declinations, 

V^-'i+^+^pT  (tan*,-  tan*,), 
which  is  equivalent  to, 


The  following  table  gives  the  value  of  the  bracketed  coefficient  in 
this  equation,  computed  with  the  argument  z=^(zl+z^),  for  an  aver- 
age condition  of  the  atmosphere,  barometer  29.00  inches,  temperature 
50°  Fahr.  In  all  ordinary  cases  the  correction  for  refraction  may 
be  found  with  sufficient  accuracy  by  multiplying  the  tabular  number, 
5,  by  the  difference  of  the  zenith  distances  of  the  two  stars,  expressed 

in  degrees, 

r=5(01-02)°.  (134) 

Since  5  is  a  positive  number,  the  correction  thus  found  will  always 
have  the  same  sign  as  the  term  zl—  z2,  measured  with  the  gradienter. 

REFRACTION    COEFFICIENTS. 


z 

s 

* 

s 

0° 
IO 

1  0 

50° 

55 

3-'04"   I 

20 

30 
40 

I'1     2 

1:1  * 

60 

65 
70 

3'9   16 

5° 

2.4 

75 

14-5 

The  use  of  the  table  is  illustrated  in  the  following  short  example 
taken  from  the  data  of  §  73: 

z         50.9 
zl  —  02       5.6 


14.0 


ACCURATE  DETERMINATIONS.  161 

Reduction  to  the  Meridian. — It  is  sometimes  conve- 
nient or  necessary  to  observe  a  star  at  some  other  instant 
than  that  of  its  meridian  passage,  and  for  this  purpose 
the  instrument  may  be  turned  out  of  the  meridian,  set 
at  an  azimuth  that  we  will  represent  by  a',  and  the  ob- 
servation made  precisely  as  before.  It  is  evident  that 
this  is  equivalent  to  observing  on  the  meridian  a  star 
whose  meridian  altitude  is  equal  to  the  altitude  of  the 
given  star  at  the  moment  of  observation,  and  whose 
declination,  therefore,  differs  from  that  of  the  latter  star 
by  the  reduction  to  the  meridian  corresponding  to  the 
azimuth  a',  (Equation  55).  In  the  reduction  of  the  ob- 
servation we  have  therefore  to  substitute  in  place  of 
the  star's  true  declination,  d,  a  corrected  declination,  <5", 
given  by  the  relation 

d"  =  d±f(a')*,      f  =  [7.9407]  cos  0  cos  h0  sec  d,     (135) 

where  a'  is  to  be  expressed  in  minutes  of  arc  and  the 
upper  sign  applies  to  a  star  between  the  zenith  and  pole, 
the  lower  sign  to  all  other  cases. 

For  the  sake  of  increased  precision  it  will  frequently 
be  advantageous  to  make  several  gradienter  pointings 
upon  a  star  in  different  azimuths,  during  the  two  or 
three  minutes  i  ^at  precede  and  follow  its  culmination, 
and,  having  first  oriented  the  instrument,  to  determine 
from  readings  of  the  horizontal  circle  the  corresponding 
azimuths  required  in  the  reduction. 

72.  Errors  of  the  Screw.  —  In  Equation  131  it  is 
tacitly  assumed  that  the  angle  moved  over  by  the  line 
of  sight  when  the  gradienter  is  turned  from  one  reading 


162  FIELD  ASTRONOMY. 

,to  another  is  strictly  proportional  to  the  amount  of 
turning  of  the  screw*  This  is,  however,  an  ideal  con- 
dition seldom  realized  in  fact,  and  if  the  capabilities 
of  the  instrument  are  to  be  fully  utilized  the  errors  of 
the  gradienter  must  be  investigated,  and  a  set  of  cor- 
rections, C,  determined,  such  that  the  angle  moved 
through  by  the  line  of  sight  when  the  gradienter  is  turned 
from  the  reading  Rl  to  R2  may  be  strictly  proportional 
to  the  difference  of  the.  corrected  readings,  R'  = 


R").  (137) 

This  calibration  of  the  screw  may  be  made  as  follows  : 
Let  some  fixed  vertical  angle,  e.g.,  the  difference  of  ele- 
vation of  two  terrestrial  points,  be  measured  upon  con- 
secutive parts  of  the  gradienter  screw,  from  the  begin- 
ning to  the  end  of  its  run,  so  that,  calling  this  angle  vt 
we  shall  have, 


(137*) 


The  second  reading  of  the  screw  in  the  first  measurement 
of  v  must  be  the  same  as  the  first  reading  in  the  second 
measurement,  etc.,  and  to  secure  this  the  gradienter 
should  not  be  touched  after  the  second  pointing,  Rlt 
has  been  made,  but  the  telescope  should  be  undamped, 
set  back  by  hand,  approximately,  upon  the  first  point 


ACCURATE  DETERMINATIONS.  163 

and  the   accurate  pointing  completed  by  means  of  the 
levelling  screws. 

From  the  mean  of  the  preceding  equations  we  obtain 

-. 


which  contains  the  three  arbitrary  quantities  k,  Cm,  C0, 
and  is  the  only  equation  that  these  quantities  are  re- 
quired to  satisfy.  We  may  therefore  impose  two  addi- 
tional relations  among  them,  and,  as  convenient  ones  for 
the  present  purpose,  we  assume  Cm  =  C0  =  c,  where  c  is 
a  constant  whose  value  we  shall,  for  the  present,  leave 

undetermined.     Representing  by  p  the  value  of  ~-  cor- 

K 

responding  to  these  assumptions, 

f=^T*>  (J39) 

and  introducing  it  into  Equations  137,  we  find  the  fol- 
lowing results : 

C0=  +c 

-R3+c,  (140) 


The  corrections  thus  derived  from  the  readings,  R, 
may  be  plotted  in  a  curve,  from  which  values  of  C  for 
all  intermediate  readings  may  be  obtained.  The  par- 
ticular value  assigned  to  c  will  have  no  influence  upon 


164 


FIELD  ASTRONOMY. 


the  shape  of  this  curve,  but  will  determine  its  position 
with  respect  to  the  axis  of  x,  and  we  may  assign  to  c 
with  advantage  a  value  that  will  make  the  entire  curve 
lie  above  the  #-axis,  i.e.,  one  that  will  make  all  the  values 
of  C  positive  quantities. 

The  following  example  represents  the  record  and 
reduction  of  a  set  of  readings  made  for  the  investigation 
of  the  errors  of  the  gradienter  of  an  engineer's  transit. 
The  quantities  in  the  column  R  are  those  directly  ob- 
served; the  column  m  gives  the  serial  number  corre- 
sponding to  that  used  in  the  above  analysis. 

Thursday,  June  7,  1900. 
Gradienter  of  Instrument  No.  386.     Observer,  P. 


nt 

R 

R0+»,f 

(R0+mj)-R 

C 

0 

o  .027 

o  .027 

o  .000 

+  31 

I 

2.037 

2  .025 

—  .012 

*9 

2 

4-045 

4.023 

—  .022 

9 

3 

6.047 

6  .022. 

-.025 

6 

4 

8.048 

8  .020 

—  .028 

3 

5 

10  .049 

10  .018 

-.031 

o 

6 

12.047 

12  .016 

-.031 

0 

7 

14.045 

14.014 

-.031 

o 

8 

16.035 

16.013 

—  .022 

9 

9 

18.022 

18  .on 

—  .on 

20 

10 

20.009 

20.009 

o  .000 

+  3i 

In  the  reduction  of  the  above  we  use 

P  =  T$  (20. 009  —  0.027)  =  1.9982. 

The  last  column,  expressed  in  thousandths  of  a  revolu- 
tion, is  obtained  by  adding  to  the  numbers  of  the  pre- 
ceding column  the  assumed  constant,  c=+  0.031.  A 
considerable  number  of  such  determinations  should  be 
made  and  the  mean  of  the  several  results  adopted  as 
definitive  corrections  to  the  gradienter  readings.  Similar 


ACCURATE  DETERMINATIONS.  165 

corrections  must  always  be  applied  where  a  high  degree 
of  precision  is  required  in  the  use  of  a  gradienter  or 
other  similar  micrometer,  e.g.,  the  eyepiece  micrometer 
of  a  zenith  telescope  or  transit,  and  particular  care  should 
be  given  to  them  in  determinations  of  k,  the  value  of  one 
revolution  of  the  gradienter  screw. 

In  a  similar  manner  the  gradienter  should  be  exam- 
ined for  periodic  errors,  i.e.,  errors  peculiar  to  a  particular 
part  of  a  turn  and  which  repeat  themselves  whenever 
the  same  part  of  the  head,  as  the  o,  comes  under  the 
index,  regardless  of  the  number  of  whole  revolutions 
at  which  the  screw  stands. 

73.  Gradienter  Latitudes.  Example.  —  We  may  now 
write  the  equation  for  zenith-telescope  latitudes  in  the 
form, 


through  which  a  value  of  the  latitude  may  be  derived 
from  each  pair  of  stars  observed,  if  k  is  known.  This 
value  of  a  revolution  of  the  screw,  k,  may  be  determined 
by  measuring  with  the  gradienter  a  known  angle,  such 
as  the  difference  of  declination  of  two  stars,  or  it  may 
be  treated  as  an  unknown  quantity  whose  value  is  to 
be  derived  from  the  latitude  observations  themselves. 
In  the  latter  case  at  least  two  pairs  of  stars,  preferably 
ten  to  twenty  pairs,  must  be  observed  for  the  deter- 
mination of  the  two  unknowns,  0  and  k,  and  these  should 
be  so  selected  that  in  one  pair  the  sum  of  the  declinations 
is  greater  than  20  and  in  the  other  pair  is  less  than  20. 
The  following  example  represents  the  observation 


166  FIELD  ASTRONOMY. 

and  reduction  of  a  single  pair  of  stars  made  with  the 
instrument  shown  in  Plate  I,  whose  errors  are  investi- 
gated in  §  72.  The  gradienter  readings  as  directly 
observed  are  given  in  the  column  marked  R,  and  in  the 
following  column  there  are  given  the  corrections  to 
these  readings  as  interpolated  from  the  table  at  p.  164. 
The  instrument  having  been  oriented  by  the  method 
of  §  32,  the  readings  of  the  horizontal  circle,  in  the  column 
H.C.,  furnish  immediately  the  azimuths,  a',  required 
for  computation  of  the  reductions  to  the  meridian,  which 
are  here  represented  by  the  letter  M.  The  stars'  merid- 
ian altitudes,  hQJ  that  are  also  needed  for  the  compu- 
tation of  these  reductions,  may  be  obtained  with  suffi- 
cient accuracy  from  the  declinations  and  the  known 
approximate  latitude  of  the  place,  43°.  The  value  of 
a  revolution  of  the  gradienter,  k,  was  known  to  be  about 
20'  30",  and  this  value  together  with  the  observed  dif- 
ference of  the  gradienter  readings  determines  zl  —  z2 
with  sufficient  precision  to  permit  the  refraction  correction 
to  be  interpolated  from  the  table  at  p.  160.  The  level 
correction,  26"  =  —  7",  is  negative  since  the  level  read- 
ings show  that  the  vertical  axis  of  the  instrument  pointed 
north  of  the  zenith,  i.e.,  in  too  great  a  latitude. 

The  declinations  of  the  stars  are  taken  from  the 
American  Ephemeris,  but  in  the  case  of  Polaris,  which 
was  observed  at  its  transit  over  the  lower  half  of  the 
meridian,  sub  polo,  the  almanac  declination  is  subtracted 
from  1 80°  in  order  to  obtain  the  distance  of  the  star 
from  the  upper  half  of  the  equator,  which  is  the  quantity 
used  in  the  analysis  and  required  in  the  reduction. 


ACCURATE  DETERMINATIONS. 


167 


Monday,  May  20,  1901. 
At  Azimuth  Stake.     Instrument  No.  389.     Observer,  C. 


Star. 

Level. 

N.          S. 

R 

Corr. 

H.  C. 

£ 

M 

Remarks. 

Polaris,  S.P. 
a  Virginis 

6.2  9.2 

8.0  7.5 

0.302 
16  .  704 

+  29 

+  13 

0             1 

*79  55 
358  21 

Off/ 

91    13    18 
-io  38   57 

it 
-6 
-5i 

Level,d  =  3" 

—  1.25 

Reduction  to  Meridian. 


Star 

Polaris 

a  Virginis 

*/  +  *,' 

80°  33'  24" 

a' 

5' 

99' 

cos  <£ 

9.863 

9.863 

2&" 

-7 

cos  /i0 

9.872 

9  .906 

sec  d 

i  .671 

o  .008 

Ref'n 

+  14 

Const. 

7.941 

7.941 

(a')2 

1.398 

3-992 

Rf  -R" 

16.386  rev. 

logM 

o-745 
-5-6 

1.710 
-51.2 

2<£  =  8o033'3 

i"  +  i6.386& 

Each  observed  pair  of  stars  furnishes  an  equation 
similar  to  the  above,  involving  0  and  k  as  unknown 
quantities,  for  which  definitive  values  are  to  be  obtained 
from  a  solution  of  all  the  available  equations.  For  illus- 
tration we  select  a  single  additional  pair  of  stars  and  its 
resulting  equation,  viz., 

20  =  91°  9'  52"- 14.671^, 
and  combining  it  with  the  one  derived  above  we  obtain, 

fc  =  i229".4,  0  =  43°  4' 38"- 

This  value  of  0  agrees  within  i"  with  the  known  latitude 
of  the  place  of  observation  and  represents  about  the 
limit  of  accuracy  attainable  with  an  engineer's  transit. 

With  the  zenith  telescope,  used  in  essentially  the 
same  manner  as  above,  a  precision  of  about  o".i  is 
attained.  See  Appendix  7,  Report  of  the  U.  S.  Coast 
and  Geodetic  Survey  for  the  Year  1897-98,  for  an  exposi- 
tion of  the  methods  employed  with  such  an  instrument. 


CHAPTER  IX. 
THE  TRANSIT  INSTRUMENT. 

74.  General  Principles.  —  Adjustments  of  the  Instru- 
ment.— If  the  celestial  meridian  were  a  visible  line  drawn 
across  the  heavens,  the  local  sidereal  time  corresponding 
to  this  meridian  might  be  determined  by  observing  the 
chronometer  time,  T,  at  which  a  star  of  known  right 
ascension,  a,  crossed  this  line.  We  should  then  have 
for  the  correction  of  the  timepiece  employed, 

AT  =  a-T. 

The  transit  instrument,  different  forms  of  which  are  shown 
in  the  Frontispiece  and  in  Plate  VI,  is  a  substitute  for 
the  visible  meridian  above  supposed.  Its  essential  parts 
are  illustrated  by  the  telescope  and  standards  of  a  large 
theodolite  firmly  mounted,  with  the  horizontal  axis  of  ro- 
tation perpendicular  to  the  plane  of  the  meridian,  i.e.,  east 
and  west,  and  level.  The  telescope  is  usually  provided 
with  several  vertical  threads  (an  odd  number  of  them), 
each  of  which,  as  seen  by  the  observer,  is  projected  against 
the  sky  as  a  background,  and  each  of  which,  when  the 
telescope  is  turned  about  the  rotation  axis,  traces  upon 
the  sky,  by  virtue  of  this  rotation,  a  circle  whose  plane  is 
perpendicular  to  the  axis.  Also,  one  or  more  horizontal 
threads  are  usually  introduced  to  mark  the  middle  points 

of  the  transit  threads. 

168 


PLATE  VI. 


A  Straight  Transit   Instrument.     Length   of  Telescope   30  inches. 
Approximate  Cost  $1000. 

[Tofacef.  168.] 


THE  TRANSIT  INSTRUMENT.  169 

A  transit  instrument  is  said  to  be  perfectly  adjusted 
when  the  circle  thus  traced  upon  the  sky  by  its  middle 
vertical  thread  coincides  with  the  local  meridian,  and  for 
such  an  instrument  it  is  evident  that  the  time  of  a  star's 
transit  over  this  thread  may  be  substituted  for  the  time 
of  its  transit  over  the  visible  meridian  above  supposed, 
and  the  chronometer  correction,  JT,  will  then  be  fur- 
nished by  the  equation  printed  above.  But  in  general 
it  cannot  be  assumed  that  these  adjustments  are  perfect, 
and  we  must  consider  them  as  so  many  possible  sources 
of  error  whose  effects  must  be  in  some  way  eliminated 
from  the  results  of  observation. 

Optical  Adjustments. — We  assume  that  great  care 
has  been  given  to  the  optical  adjustment  of  the  instru- 
ment, so  that  both  the  transit  threads  and  the  star  are 
sharply  defined  and  distinctly  seen.  For  this  purpose 
the  eyepiece  should  first  be  so  set  that  the  threads  appear 
black  and  distinct,  and  threads  and  eyepiece  should 
then  be  moved  in  or  out  together  until  a  star,  preferably 
a  double  star,  presents  a  clear  image  without  trace  of 
fuzziness,  projecting  rays,  or  stray  light.  This  last  ad- 
justment may  be  a  little  more  accurately  made  by  cov- 
ering the  upper  half  of  the  telescope  objective  with  card- 
board or  paper  and  making  an  accurate  pointing  of  the 
horizontal  thread  upon  a  circumpolar  star  near  cul- 
mination. Having  made  a  satisfactory  pointing,  quickly 
shift  the  card  so  as  to  cover  the  lower  half  of  the  objective 
and  leave  free  the  upper  part,  when,  if  the  threads  are 
not  properly  adjusted  with  respect  to  the  objective, 
there  will  be  a  slight  vertical  displacement  of  the  star 


170  FIELD  ASTRONOMY. 

with  respect  to  the  thread,  and  this  must  be  corrected 
by  further  adjustment. 

Vertically  of  Threads. — To  make  the  threads  perpen- 
dicular to  the  rotation  axis,  point  the  telescope  at  a  ter- 
restrial mark,  and  turning  the  telescope  in  altitude 
with  the  slow-motion  screw,  note  whether  the  mark  in 
its  apparent  motion  up  and  down  the  field  of  view  runs 
exactly  along  the  thread.  Any  outstanding  error  in 
this  adjustment  may  be  removed  by  slightly  rotating 
in  its  own  plane  the  collar  which  carries  the  threads ;  but 
a  small  error  here  may  be  rendered  harmless  by  always 
pointing  the  telescope,  at  the  times  of  observation,  so 
that  the  stars  cross  the  same  part  of  the  field,  e.g.,  be- 
tween the  parallel  horizontal  threads. 

The  principal  errors  of  adjustment  that  remain  to 
be  considered  in  connection  with  the  use  of  a  transit 
instrument  are  three  in  number,  viz. :  The  azimuth  error, 
a,  is  the  angular  amount  by  which  the  rotation  axis 
deviates  to  the  south  of  due  west.  The  level  error,  6,  is 
the  angle  of  elevation  of  the  rotation  axis  above  the 
western  horizon.  The  collimation  error,  c,  is  the  amount 
by  which  the  angle  between  the  line  of  sight  and  the 
west  half  of  the  rotation  axis  exceeds  90°.  The  line  of 
sight  as  here  used  means  the  imaginary  line  passed 
through  the  optical  centre  of  the  objective  and  the  mid- 
dle transit  thread,  or  through  the  mean  of  a  group  of 
transit  threads. 

75.  Theory  of  the  Instrument. — To  determine  the  rela- 
tion of  these  several  instrumental  errors  to  the  time,  T, 
at  which  a  star  will  pass  behind  a  given  transit  thread 


THE  TRANSIT  INSTRUMENT.  171 

we  have  recourse  to  Fig.  13,  which  represents  a  projec- 
tion of  the  celestial  sphere  upon  the  plane  of  the  horizon. 
Z  is  the  projection  of  the  zenith,  P  of  the  pole,  H  of  the 
point  in  which  the  rotation  axis,  produced  toward  the 
west,  intersects  the  celestial  sphere,  and  5  is  the  projec- 
tion of  a  star  observed  at  the  moment  of  its  transit  over 


FIG.  13. — The  Transit  Instrument. 

a  thread  whose  angular  distance  from  H  is  measured 
by  the  arc  90°  +  c.  From  the  definitions  given  above,  c 
represents  the  collimation  of  the  particular  thread  in 
question,  and  similarly  b  and  a,  in  the  figure,  are  the 
level  and  azimuth  errors  above  defined.  The  symbol 
r  of  the  figure  represents  the  hour  angle  of  the  star, 
reckoned  toward  the  east,  d  is  the  star's  declination, 
90°-  K  is  the  arc  HM,  and  A  is  the  distance  of  the  star 
from  the  meridian  measured  along  HS.  This  latter  arc 


172  FIELD  ASTRONOMY. 

must  not  be  confounded  with  the  diurnal  path  of  the 
star;  the  one  is  an  arc  of  a  great  circle  defined  by  the 
points  H  and  5,  while  the  other  is  a  small  circle  having 
its  pole  at  P.  Note  that  in  all  cases  the  symbols  here 
defined  represent  the  actual  magnitudes  of  the  arcs 
and  angles  on  the  sphere,  and  not  of  their  projections  on 
the  plane  of  the  horizon. 

From  the  spherical  triangle  PMS  we  obtain  the  rela- 
tion, 

cos  d  sin  T  =  sin  X  sin  $,  (143) 

and  from  the  triangle  ZHM  we  find, 

sin  K  =  sin  b  cos  £  +  cos  b  sin  £  sin  a.  (144) 

These  equations  may  be  greatly  simplified  by  substi- 
tuting arcs  in  place  of  sines  whenever  the  quantities  a 
and  b  are  so  small  that  their  cubes  and  higher  powers 
may  be  neglected,  and  we  shall  therefore  assume  that 
we  have  to  deal  with  an  approximately  adjusted  instru- 
ment, in  which  neither  of  these  quantities  much  exceeds 
10'.  On  this  supposition  the  point  H  is  so  nearly  the 
pole  of  the  meridian,  PZM,  that  we  may  put  sin  0  =  i 
and  C  =  0-£,  where  0  denotes  the  latitude  of  the  place 
of  observation,  and  our  equations  now  take  the  form, 

T  =  X  sec  d, 
K  =  b  cos  (<t>-d)  +a  sin  (<t>-d). 

From  the  figure  we  have  the  relation, 


THE  TRANSIT  INSTRUMENT.  173 

and  eliminating  A  between  these  equations  we  find, 
T  =  sin  (0—5)  sec  5.  a  -f  cos  (0—5)  sec  5.  6  +  sec  d  .c.  (146) 

Since  T  is  an  east  hour  angle,  we  have  also,  in  terms  of  the 
observed  time,  the  chronometer  correction,  and  the  star's 
right  ascension, 

r  +  jr  =  a-r,  (147) 

from  which  we  obtain,  by  the  elimination  of  T,  Mayer's 
equation  of  the  transit  instrument, 


0-5)  sec  d  .  a 
+  cos  (0-5)  sec  d  .  6  +  sec  d  .  c,     (148) 

or,  as  it  is  usually  written, 

a-T  =  AT  +  Aa  +  Bb  +  Cc,  (149) 

where  the  capital  letters  are  introduced  as  abbreviations 
for  the  coefficients  given  above,  i.e., 

yl=sin(0-5)  sec  5,  £=cos(0-5)  sec  5,  C  =  sec5.  (150) 

Since  a,  T,  and  AT  are  expressed  in  time  (hours,  minutes, 
and  seconds),  it  is  customary  in  connection  with  this 
equation  to  express  a,  6,  and  c  in  seconds  of  time. 

76.  Discussion  of  Mayer's  Equation.  —  The  coefficients 
A,  B,  C  are  called  transit  factors,  and  when  many  obser- 
vations are  to  be  made  in  the  same  latitude,  0,  as  at  an 
observatory,  it  is  customary  to  tabulate  their  values 
with  the  declination  as  argument,  and  to  interpolate 
from  these  tables  the  values  of  the  factors  corresponding 
to  the  particular  stars  observed.  In  the  U.  S.  Coast  and 
Geodetic  Survey  Report  for  the  year  1880  there  may  be 


174  FIELD  ASTRONOMY. 

found  extensive  tables  of  this  kind  for  different  latitudes 
covering  the  whole  extent  of  the  United  States. 

In  the  use  of  such  tables^  the  following  distinction 
must  be  carefully  observed:  Every  star  whose  distance 
from  the  pole  is  less  than  the  latitude  remains  continu- 
ously above  the  horizon  throughout  the  twenty-four 
hours,  and  during  this  period  crosses  the  meridian  twice, 
once  above  the  pole,  e.g.,  between  the  pole  and  zenith, 
and  once  below  the  pole,  e.g.,  between  the  pole  and  the 
northern  horizon.  The  latter  transit  is  usually  desig- 
nated sub  polo,  and  from  Fig.  13,  where  5'  represents 
the  star  5  near  its  transit  sub  .polo,  it  may  be  seen  that 
its  coordinates  at  this  transit  will  be  obtained  by  sub- 
stituting in  place  of  the  a  and  90°  -d,  corresponding 
to  5,  i2h  +  a  and  —  (90°—  £).  When  these  values  are 
introduced  into  Mayer's  equation  it  becomes,  for  stars 
observed  sub  polo, 

i2h  +  a-  T  =  4T  +  A'a  +  B'b  +  C'c,  (151) 

where  the  new  transit  factors  have  the  following  values  : 
A'  =sin  (0+  d)  sec  d,     Ef  =cos  (0+  d)  sec  d, 


As  an  exercise  in  analysis  the  student  may  show  that 
the  transit  factors  for  a  star  above  and  below  the  pole  are 
connected  by  the  relations, 

o.     (153) 


Use  these  equations  to  derive  A',  B',  C'  from  the  tabulated 
values  of  A,  B,  and  C. 

From  a  consideration  of  the  trigonometric  functions 


THE  TRANSIT  INSTRUMENT.  175 

that  enter  into  the  transit  factors  the  algebraic  signs  of 
these  factors  are  found  to  be  as  follows  for  places  in  the 
northern  hemisphere: 

Factor.  A  B  C 

South  of  Zenith +          +          +  - 

Zenith  to  Pole -          -f-          +  - 

Below  Pole +  -  + 

Note  that  in  every  case  the  transit  factors  for  a  given 
star  have  opposite  signs  above  and  below  the  pole,  and 
compare  with  this  statement  the  fact  that  stars  on  oppo- 
site sides  of  the  pole  move  in  opposite  directions,  east 
to  west  above  pole  and  west  to  east  below  pole. 

Query. — The  above  relations  of  sign  are  for  a  place 
in  north  latitude.  How  must  they  be  changed  to  adapt 
them  to  a  place  south  of  the  terrestrial  equator  ? 

In  explanation  of  the  double  set  of  signs  given  above 
for  C,  we  recall  what  was  shown  in  §  50,  that  a  reversal 
of  the  instrument  changes  the  sign  of  the  collimation 
constant,  c;i.e.,  90°  —  c  is  substituted  for  the  90°  -\-c  of 
Fig.  13,  by  lifting  the  axis  out  of  the  wyes  and  replacing 
it,  turned  end  for  end.  It  is  customary  to  ignore  this 
change  of  sign  in  c,  and  to  represent  its  effect  in  Mayer's 
equation  by  changing  the  algebraic  sign  of  C  when  the 
instrument  is  reversed;  e.g., 

For  Circle  W C(  +  c)  =  (  +  C)c 

For  Circle  E C(-c)  =  (-C)c 

The  collimation  constant,  c,  may  be  either  positive  or 
negative,  depending  upon  the  adjustment  of  the  instru- 
ment ;  but  it  retains  the  same  sign  in  both  positions  of  the 


176  FIELD  ASTRONOMY. 

circle,  while  the  collimation  factor,  C,  is  positive  (above 
pole)  when  the  circle  end  of  the  axis  points  west,  negative 
when  it  points  east. 

77.  Choice  of  Stars.  —  In  ;the  right-hand  member  of 
Mayer's  equation,  as  printed  on  page  173,  there  are  in- 
volved four  unknown  quantities,  JT,  a,  b,  and  c,  one  of 
which,  b,  the  inclination  of  the  axis  to  the  plane  of  the 
horizon,  is  always  to  be  determined  by  some  mechanical 
method,  e.g.,  the  use  of  a  spirit-level.  The  collimation 
constant,  c,  may  also  be  determined  mechanically  (see 
§  84),  but  for  the  present  we  shall  assume  that  this  has 
not  been  done  and  that  the  instrumental  constants  a 
and  c,  as  well  as  the  clock  correction  AT,  are  to  be  deter- 
mined from  observations  of  stars.  Since  there  are  three 
quantities  to  be  thus  determined,  there  must  be  at  least 
three  observations,  and  it  is  practically  convenient  to 
make  four  the  minimum  number  instead  of  three;  ob- 
serving two  stars  Circle  E.  and  two  Circle  W.  for  the  sake 
of  a  good  determination  of  the  collimation,  c,  through 
the  reversal  of  the  instrument.  The  stars  thus  chosen 
should  not  all  lie  on  the  same  side  of  the  zenith,  but 
should  be  distributed  on  both  sides,  so  as  to  make  the 
sum  of  their  azimuth  factors  as  small  as  possible.  When 
I A  =o,  the  effect  of  the  azimuth  error,  a,  is  completely 
eliminated,  and  a  nearly  complete  elimination  may 
usually  be  obtained  by  care  in  the  selection  of  stars.  In 
the  example  of  §  78  this  condition  is  approximately 
satisfied  by  the  four  stars  marked  a',  b',  d' ,  e' ,  and  the 
student  after  tracing  through  the  reduction  there  given, 
should  note  that  if  the  azimuth  star,  i  H.  Draco.,  were 


THE  TRANSIT  INSTRUMENT.  177 

dropped  and  the  azimuth  error  entirely  ignored,  the 
resulting  value  of  AT  would  be  substantially  the  same 
as  is  obtained  when  the  azimuth  error  is  taken  into 
account.  In  this  case,  therefore,  an  accurate  deter- 
mination of  a  is  of  little  consequence. 

78.  Example. — Ordinary  Determination  of  Time. — The 
following  example,  taken  from  the  time  service  of  the 
Washburn  Observatory,  0  =  43°  4' 3 7",  illustrates  the 
record  and  reduction  of  a  set  of  transit  observations.  In 
addition  to  the  date  and  the  measured  inclination,  6, 
of  the  horizontal  axis,  given  in  the  column  of  Constants 
for  the  two  positions  of  the  instrument,  Circle  W.  and 
Circle  E.,  the  observed  data  are  contained  in  the  three 
columns  marked,  at  the  foot,  with  Roman  numerals, 
I,  II,  III.  The  observed  times  of  transit  given  in  III 
are  each  the  mean  of  the  observed  times  of  transit  of 
the  given  star  over  1 5  threads,  and  in  the  reduction  the 
collimation  constant,  c,  is  assumed  to  refer  to  the  mean 
of  these  threads  instead  of  to  the  middle  thread.  Note 
that  this  particular  convention  with  regard  to  c  can  be 
adopted  only  when  each  star  is  observed  over  precisely 
the  same  set  of  threads  as  every  other  star.  The  failure 
to  observe  a  single  star  at  its  transit  over  one  of  the 
threads  will  require  either  the  rejection  of  the  transits 
of  other  stars  observed  at  this  thread,  or  a  determination 
of  thread  intervals  and  a  * '  reduction  to  the  mean  thread  " 
for  which  reference  may  be  made  to  Appendix  7,  U.  S. 
Coast  and  Geodetic  Survey,  Annual  Report  for  1897-98. 

The  remaining  columns  are  marked  with  Arabic  nu- 
merals, showing  the  order  in  which  they  are  reached  in 


178  FIELD  ASTRONOMY. 

the  computation.  Of  these  columns  i  and  2  are  obtained 
from  the  almanac  (in  this  case  the  Berliner  Astronomisches 
Jahrbuch,  plus  the  corrections  given  in  Astronomische 
Nachrichten,  No.  3508).  The  declinations  are  taken  to 
the  nearest  minute  only,  while  the  right  ascensions  are 
accurately  interpolated  for  the  instant  of  the  star's  transit 
over  the  local  meridian,  i.e.,  0.3  day  after  their  transit 
over  the  meridian  for  which  the  almanac  is  constructed. 
The  third  star,  being  observed  sub  polo  (and  before  mid- 
night), was  observed  half  a  day  before  its  transit  over 
the  local  upper  meridian,  and  its  right  ascension  is  there- 
fore interpolated  for  an  instant  0.2  day  before  its  transit 
©ver  the  Berlin  meridian. 

The  transit  factors  contained  in  columns  4,  5,  and  6  were  inter- 
polated from  tables  of  such  factors,  and  the  products  contained  in 
columns  7  and  8  were  next  filled  in  by  the  use  of  Crelle's  multiplication 
tables.  It  may  be  noted  that  the  effect  of  diurnal  aberration  shown 
in  column  7  has  already  been  found  (§  27)  to  be  — os.o2i  cos  <£  sec  8, 
which,  for  the  given  latitude,  is  equal  to  —  os.oi5  C,  and  the  collimation 
factor  C  was  employed  in  computing  the  correction.  These  corrections 
were  next  added  mentally  to  the  numbers  contained  in  III,  and  the 
resulting  times  subtracted  from  the  right  ascensions  in  i ,  thus  giving 
the  absolute  terms  of  the  equations  numbered  9.  The  first  members 
of  these  equations,  3,  4,  6,  are  obviously  derived  from  Mayer's  equa- 
tion. 

We  have  now  five  equations  involving  only  three  unknown  quan- 
tities and  presenting,  therefore,  a  case  for  the  application  of  the  Method 
of  Least  Squares.  A  rigorous  solution  by  that  method  furnishes  the 
following  values  of  the  quantities  sought: 

JT=-f  2m57s.oio,  a= +08.858,  c=+os.g66. 
But  such  a  solution  is  rather  laborious,  and  a  simple  method  of  obtain- 
ing approximately  accurate  results  is  indicated  under  the  heading 
Solution,  where  the  symbols  at  the  left  indicate  the  manner  in  which 
the  successive  equations  are  derived.  Equation  kr  is  derived  from 
i'  by  dividing  through  by  the  coefficient  of  c,  and  /'  is  similarly  de- 
rived from  hf,  using  the  coefficient  of  4T,  as  divisor  and  substituting 
in  place  of  c  its  value  given  by  kf '.  Equation  m'  is  obtained  from  c' 
by  substituting  in  place  of  AT  and  c  their  values  as  given  in  kf  and  /'. 


THE  TRANSIT  INSTRUMENT.  179 

The  value  of  a  furnished  by  this  equation  when  substituted  in  k'  and 
/'  gives  definitive  values  of  AT  and  c,  all  of  which  are  entered  in 
the  column  of  Constants. 

By  means  of  these  values  of  a  and  c,  columns  12  and  13  are  filled 
up  and  the  sum  of  the  corrections  contained  in  columns  7,  8,  12,  13, 
is  entered  in  14  and  added  to  the  corresponding  numbers  in  III,  thus 
furnishing  the  corrected  times  contained  in  15.  Only  the  seconds 
are  entered  here,  since  the  minutes  remain  unchanged.  The  indi- 
vidual values  of  the  clock  correction  contained  in  16  are  now  obtained 
by  subtracting  15  from  i,  and  their  agreement,  one  with  another,  is 
a  check  upon  the  accuracy  of  the  entire  work,  both  observations  and 
computations.  For  the  sake  of  this  check  it  is  better  to  proceed  as 
is  here  done  than  to  rely  upon  the  value  of  AT  furnished  by  the  solu- 
tion of  the  equations.  The  numerical  work  here  shown  is  greatly 
facilitated  by  the  use  of  a  slide-rule  or  an  extended  multiplication 
table  such  as  that  of  Crelle. 

It  may  readily  be  seen  from  the  course  of  the  above 
solution  that  the  collimation,  c,  is  obtained  from  the 
four  observations  marked  a',  b',  d',  e' ,  while  the  azimuth, 
a,  is  furnished  by  the  third  observation.  A  star  near 
the  pole,  like  i  H.  Draco.,  is  introduced  into  the  observ- 
ing programme  solely  to  determine  a,  and  with  refer- 
ence to  this  use  it  is  called  an  azimuth  star,  while  the 
others  are  known  as  clock  stars,  since  it  is  they  that  deter- 
mine the  value  of  AT.  As  there  is  always  a  possibility 
of  disturbing  the  azimuth,  i.e.,  changing  a  in  the  act  of 
reversing  the  instrument,  there  should,  in  all  strictness, 
be  two  values  of  a  determined,  one  for  Circle  W.  as  well 
as  the  one  above  found  from  the  observation  of  a  star 
Circle  E. ;  but  in  the  present  case  it  may  readily  be  seen 
that  there  was  no  such  disturbance,  since  the  value  of  a 
for  Circle  E.  brings  into  perfect  agreement  the  values 
of  AT  furnished  by  the  two  stars  observed  Circle  W., 
although  their  azimuth  factors  are  widely  different. 

Whenever  necessary,  introduce  into  the  solution  two 
azimuths,  one  for  each  position  of  the  instrument,  as 


180 


FIELD  ASTRONOMY. 


8    S 

n     M 
2    o 


c/3 


Constants. 

CN     ^    IONO     HI 
.O    Hi  00    ON  O 
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THE  TRANSIT  INSTRUMENT.  181 

unknown  quantities.     It  is  not  necessary  to  introduce 
two  collimations. 

79.  Methods  of  Observation. — A  clock  or  chronometer 
is  an  indispensable  auxiliary  to  a  transit  instrument,  and 
an  observation  with  the  latter  consists  in  determining, 
as  accurately  as  may  be,  the  chronometer  time  at  which 
a  particular  star  transits  over  one  or  more  of  the  threads. 
In  the  best  astronomical  practice  a  recording  machine, 
called  a  chronograph,  is  used  in  this  connection,  but  we 
shall  here  suppose  the  observer  not  to  be  provided  with 
a  chronograph  and  constrained,  therefore,  to  use  the 
older  method  of  observing  by  eye  and  ear.  In  this 
method  the  observer  picks  up  the  beat  of  the  chronometer, 
i.e.,  counts  mentally  the  tick  corresponding  to  each  suc- 
cessive second,  i,  2,  3,  4,  etc.,  and  while  thus  counting 
looks  into  the  telescope  and  watches  the  progress  of  the 
star  across  the  field  of  view,  noting  its  position  at  the 
instant  of  each  counted  beat.  If,  by  any  chance,  the 
star  should  appear  exactly  behind  a  thread  at  the  instant 
when  the  counted  beat  was  26,  the  time  of  transit  over 
this  thread  would  be  recorded  26.0  seconds,  and  the  cor- 
responding hour  and  minute  subsequently  determined 
by  looking  at  the  face  of  the  chronom- 
eter. It  will  usually  happen,  however, 
that  the  star  passes  behind  the  thread 
between  two  chronometer  beats  instead 
of  simultaneously  with  one  of  them, 
somewhat  as  shown  in  Fig.  14,  where  FlG>  i4._Transits 
there  is  indicated  the  position  of  the  star  by  Eye  and  Ean 
with  respect  to  the  thread  at  26*  and  at  2ys,  as  noted 


182  FIELD  ASTRONOMY. 

and  temporarily  remembered  by  the  observer.  From 
the  manner  in  which  the  thread  divides  the  space  between 
the  two  star  images  it.  is  evident  that  the  actual  transit 

'  .» 

over  the  thread  occurred  at  26.4s,  and  it  should  be  so 
recorded.  The  fraction  of  a  second  depends  upon  the 
observer's  estimation  (an  estimate  of  space  seen  in  the 
telescope  and  not  time  as  counted  by  the  ear),  and  a 
skilled  observer  should  be  able  to  follow  a  star  in  its 
progress  across  the  field  of  view,  observing  and  recording 
to  the  nearest  tenth  of  a  second  the  times  of  transit  over 
as  many  threads  as  may  be  desired,  without  taking  the 
eye  from  the  telescope  during  the  process.  He  should, 
while  watching  the  star,  give  no  heed  to  the  hour  and 
minute,  but  concentrate  attention  upon  the  seconds 
and  fractions  of  a  second,  until  the  transit  over  the  last 
thread  has  been  recorded;  then,  still  counting  seconds, 
let  him  look  back  at  the  face  of  the  chronometer  and 
note  if  the  time  there  shown  by  the  seconds  hand  agrees 
with  his  count.  This  is  called  checking  the  beat,  and 
if  it  checks  properly,  the  minute  and  hour  corresponding 
to  the  last  observation  should  be  written  down  as  a  part 
of  the  record. 

80.  Precision  of  the  Results.  —  By  the  method  above 
outlined  a  skilled  observer  may,  from  the  mean  of  several 
threads,  determine  the  time  of  a  star's  transit  within  very 
small  limits  of  error;  e.g.,  there  is  found  for  the  probable 
error  of  a  transit  of  a  single  star  over  the  mean  of  from 
10  to  15  threads,  some  os.o2  or  os.o3.  But  this  apparent 
precision  is  in  some  degree  fallacious,  for  most  observers 
possess  individual  peculiarities,  called  personal  equation, 


THE   TRANSIT  INSTRUMENT.  183 

by  which  they  tend  to  observe  all  stars  either  too  soon 
or  too  late,  by  a  nearly  constant  amount,  and  the  proba- 
ble error  of  a  transit  based  upon  the  agreement  of  indi- 
vidual results,  one  with  another,  furnishes  no  indication 
of  the  presence  or  magnitude  of  this  constant  personal 
error. 

Closely  related  to  the  precision  attainable  in  esti- 
mating the  times  of  transit  of  a  star  over  the  threads  of 
an  instrument,  is  the  degree  of  accordance  to  be  expected 
among  the  values  of  AT  furnished  by  the  several  stars 
composing  a  set,  such  as  that  of  the  illustrative  example 
of  §  78.  The  range  of  values  there  exhibited,  while 
smaller  than  is  to  be  expected  from  a  beginner,  may  be 
regarded  as  fairly  typical  of  the  results  to  be  obtained  by 
an  experienced  observer  provided  with  a  good  instru- 
ment. See  in  this  connection  the  example  of  §  82,  where 
the  results  show  an  even  closer  but  by  no  means  abnor- 
mal agreement. 

81.  Personal  Equation.  —  The  personal  equation,  al- 
though a  real  and  oftentimes  a  considerable  source  of 
error,  is,  however,  of  small  consequence  save  where  the 
observations  of  different  persons  are  to  be  combined,  one 
with  another,  as  in  a  determination  of  longitude.  In  such 
cases,  however,  the  problem  of  personal  equation  must 
be  met  and  seriously  dealt  with,  and  various  devices  have 
been  employed  for  this  purpose;  e.g. :  (i)  An  exchange  of 
observers  at  the  middle  of  the  work  in  question,  so  that 
its  first  half  may  be  affected  with  the  personal  error  in 
one  direction  and  the  second  half  in  the  opposite  direc- 
tion, thus  eliminating  this  influence  from  the  mean. 


184  FIELD  ASTRONOMY. 

(2)  The  determination  of  the  exact  amount  of  the  per- 
sonal equation  for  each  observer,  by  means  of  so-called 
personal-equation  machines,  is  sometimes  attempted; 
but  at  present  the  best  device  for  the  elimination  of 
personal  equation  seems  to  be:  (3)  The  Repsold  Transit 
Micrometer,  an  apparatus  in  whose  use  the  methods  of 
observing  above  set  forth,  §  79,  are  completely  aban- 
doned, and  as  a  substitute  for  them  the  observer,  while 
looking  into  the  telescope,  seeks  to  keep  the  image  of  a 
star,  as  it  moves  across  the  field,  constantly  covered  by 
a  micrometer  thread,  which  he  manipulates  with  his  rin- 
gers and  which  is  so  connected  with  a  chronograph  as 
to  give  an  automatic  record  of  the  star  transits.  The 
experience  of  the  Prussian  Geodetic  Institute  indicates 
that  in  this  mode  of  observing,  personal  differences 
between  observers  are  nearly  annihilated. 

82.  Reversal  of  the  Instrument  upon  Each  Star. — A 
method  of  using  a  transit  instrument  introduced  into 
general  practice  in  connection  with  the  transit  microm- 
eter, but  which  may  be  equally  well  applied  with  the 
ordinary  chronographic  or  eye-and-ear  methods,  con- 
sists in  noting  the  time  of  transit  of  a  star  over  a  group 
of  threads  placed  at  some  little  distance  from  the  centre 
of  the  field,  then,  after  quickly  reversing  the  instrument, 
to  observe  the  same  star  again  on  the  same  threads  in 
their  new  position.  It  is  obvious  that  the  effect  of 
collimation  is  thus  completely  eliminated  from  the  mean 
of  the  observations  on  each  thread  and  therefore  from 
the  general  mean  of  the  observed  times.  This  elimina- 
tion, while  an  important  advantage  of  this  mode  of  ob- 


THE  TRANSIT  INSTRUMENT.  185 

serving,  is  far  from  being  the  only  one,  and  a  considerable 
number  of  sources  of  error  which  have  not  been  con- 
sidered above,  but  which  are  dealt  with  at  length  in  the 
larger  treatises,  such  as  Chauvenet,  Spherical  and  Practi- 
cal Astronomy,  are  equally  eliminated  by  the  reversal; 
e.g.,  inequality  of  pivots,  flexure,  thread  intervals,  and 
the  disturbance  of  the  spirit-level  incident  to  reversing 
it  upon  the  axis.  When  the  telescope  is  reversed  upon 
every  star  a  hanging  level  may  be  allowed  to  remain 
upon  the  axis  without  ever  being  reversed,  since  the 
level  readings  in  the  two  positions  of  the  axis  then  give 
its  mean  inclination,  which  is  the  datum  required  for  the 
reduction  of  the  star  transits. 

Whenever  it  can  be  employed  the  method  of  reversal 
upon  every  star  is  to  be  preferred  to  the  older  method 
illustrated  in  the  preceding  example,  but  it  requires 
special  facilities  for  quick  reversal  of  the  instrument 
without  disturbing  its  azimuth,  and  these  are  not  always 
present. 

The  following  is  an  example  of  the  record  and  reduc- 
tion of  such  a  series  of  observations,  made  with  the  same 
instrument  and  arranged  in  nearly  the  same  manner  as 
the  example  on  p.  180.  Each  star  was  observed  on  five 
threads  in  each  position  of  the  instrument,  and  a  value 
of  the  level  constant,  6,  was  determined  for  each  star 
from  readings  of  the  hanging  level,  taker;  immediately 
before  or  after  the  observed  transits  in  leach  position 
of  the  circle,  the  level  remaining  unreversed  upon  the 

axis  during  the  entire  set  of  observations. 

,,  - . 


186 


FIELD  ASTRONOMY. 


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THE  TRANSIT  INSTRUMENT.  187 

83.  Determination  of  Azimuth  with  a  Transit  Instru- 
ment.— Let  the  line  of  sight  of  a  transit  instrument  be 
supposed  directed  accurately  upon  some  terrestrial  mark, 
and  the  telescope  then  turned  up  to  the  sky  and  the 
time  of  a  star's  transit  over  the  line  of  sight  observed. 
From  this  observed  time  the  hour  angle  of  the  star  may 
be  derived,  and  this  hour  angle,  in  connection  with  the 
known  declination  and  latitude,  will  determine  the  star's 
azimuth  at  the  instant  of  observation.  If  there  are  no 
instrumental  errors  present,  this  computed  azimuth 
will  be  the  true  azimuth  of  the  terrestrial  mark  at  which 
the  line  of  sight  was  originally  directed. 

This  simple  method  of  determining  azimuth  requires 
some  modifications  on  account  of  instrumental  errors,  but 
when  these  are  duly  taken  into  account  and  a  proper 
selection  of  stars  and  mark  is  made,  the  method  ranks 
as  the  best  of  all  known  ones  for  azimuth  determination. 
The  star  to  be  observed  should  be  very  near  the  pole, 
usually  Polaris,  and  if  the  chronometer  correction,  AT, 
is .  accurately  known,  the  observation  may  be  made  at 
any  convenient  time,  e.g.,  the  time  at  which  the  star 
stands  directly  above  a  mark  already  established.  If 
the  chronometer  correction  is  not  well  determined,  the 
observation  should  be  made  when  the  star  is  near  elonga- 
tion, since  the  effect  upon  the  computed  azimuth  of 
an  error  in  the  assumed  AT  is  then  a  minimum.  But 
this  latter  procedure  requires  the  establishment  of  a 
special  mark  whose  azimuth  shall  be  very  approximately 
equal  to  that  of  the  star  when  at  elongation,  and  it  will 
often  be  more  convenient  to  determine  the  time  with 


188  FIELD  ASTRONOMY. 

the  required  accuracy,  e.g.,  one  tenth  of  a  second,  and 
thus  obtain  more  freedom  in  the  choice  of  a  mark. 

A  transit  instrument  of  the  better  class  is  usually 
provided  with  an  eyepiece  micrometer,  i.e.,  one  or 
more  threads  parallel  to  the  fixed  transit  threads,  but 
capable  of  being  moved  to  and  fro  in  the  field  of  view 
by  a  screw  whose  axis  is  parallel  to  the  rotation  axis 
of  the  instrument.  This  screw  is  provided  with  a  grad- 
uated head  whose  readings  indicate  the  successive  posi- 
tions of  the  thread  and  measure  the  amount  of  its  motion 
between  consecutive  pointings  upon  the  star  and  mark. 
When  such  a  micrometer  is  present,  transits  of  the  star 
may  be  observed  over  its  threads  as  long  as  the  star 
remains  within  the  field  of  view,  and  many  comparisons 
between  star  and  mark  may  be  substituted  for  the  single 
one  above  supposed.  The  instrument  should  be  re- 
versed at  least  once  during  these  observations,  and  the 
inclination  of  its  axis,  b,  must  be  carefully  determined 
since,  as  will  appear  later,  the  level  error  has  an  important 
effect  upon  the  azimuth. 

84.  Theory  of  the  Method.  —  To  derive  from  the 
micrometer  readings  upon  star  and  mark  the  difference 
of  their  respective  azimuths  we  have  recourse  to  Fig.  15, 
which  represents  a  projection  of  the  celestial  sphere  upon 
the  plane  of  the  horizon.  5  and  M  represent  respectively 
the  star  and  the  mark,  Z  is  the  zenith,  and  the  spherical 
angle  SZM  is  the  required  difference  of  azimuth.  Let 
H  be  the  point  in  which  the  rotation  axis  of  the  instru- 
ment, produced  toward  the  west,  meets  the  celestial 
sphere,  and  the  arcs  90°  —  b,  90°  +  c,  will  then  have  the 


THE  TRANSIT  INSTRUMENT. 


189 


same  significance  as  in  Fig.   13.     The  spherical  angles 
HZS  and  HZM  are  represented  by  the  symbols  90 


FIG.  15. — Azimuth  with  a  Transit  Instrument. 

and  90°  +  w',  and  the  zenith  distance  of  the  star,  ZS,  by  z. 
From  the  triangle  H ZS  we  obtain, 

cos  (90°  +  c)  =  cos  (90°  —  b)  cos  z 

+  sin  (90°  —  b)  sin  z  cos  (90°  -\-w),     (154) 

which,  when  b  and  c  do  not  much  exceed  10',  is  equivalent 
to, 

c  +  b  cos  z=w  sin  z.  (I55) 

In  this  equation  we  substitute  goQ  —  h  in  place  of  z  and 
put  sec  h  =  i  +  <r,  and  it  becomes, 

W  =  C  +  C<T-{-&  tan  /*.  (156) 

From  the  triangle  ZHM  we  find  in  a  similar  manner  for 
the  mark, 

dS7) 


UNIVERSITY 


190  FIELD  ASTRONOMY. 

When  the  mark,  M,  is  in  or  very  near  the  horizon,  as  it 
should  be,  the  last  two  terms  of  Equation  157  vanish 
and  we  obtain  by  subtracting  it  from  Equation  156, 


A-A'  =w-wf  =c-c'  +  <7(7  +  6tan  h.         (158) 

Let  k  represent  the  angular  equivalent  (value)  of  one 
revolution  of  the  micrometer  screw,  R  the  reading  of  the 
screw-head  corresponding  to  any  position  of  the  movable 
thread,  and  RQ  the  particular  reading  at  which  the  angle 
between  the  rotation  axis  of  the  instrument  and  the  line 
of  sight  defined  by  the  thread  equals  90°,  i.e.,  RQ  is  the 
reading  corresponding  to  c  =  o.  For  any  other  position 
of  the  threads  corresponding  to  the  reading  R  we  shall 
have 

c=±k(R-RJ,  (159) 

where  the  ambiguous  sign  depends  upon  the  position  of 
the  instrument,  whether  Circle  W.  or  Circle  E.  For  any 
given  instrument  it  is  well  to  determine,  by  trial,  once 
for  all,  in  which  of  these  positions  the  readings  of  the 
micrometer  head  continuously  diminish  as  the  microm- 
eter thread  is  made  to  follow  the  diurnal  motion  of  a 
star  near  upper  culmination,  and,  with  reference  to  the 
sign  in  Equation  159,  designate  this  as  the  positive,  the 
other  as  the  negative,  position  of  the  instrument. 

Corresponding  to  the  positive  and  negative  positions, 
respectively,  let  R^  and  R2  be  readings  of  the  micrometer 
head  when  the  micrometer  thread  is  pointed  upon  the 
same  fixed  object,  e.g.,  the  mark  whose  azimuth  is  to 


THE  TRANSIT  INSTRUMENT.  191 

be  determined;  we  shall  then  have  as  the  distance  of 
this  object  from  the  line  of  no  collimation, 

Positive  Position,       s=  -\-k  (Rl  —  RQ), 
Negative  Position,     s=-k(R2-RQ)t 
from  which  we  readily  obtain, 

2s=+k(R1-R2), 
2R0  =  Rl  +  R2. 

The  first  of  these  equations  determines  the  distance,  st 
of  the  terrestrial  mark  from  the  collimation  axis  of  the 
instrument,  and  it  should  be  used  to  make  the  distance 
of  the  azimuth  mark  small,  by  properly  placing  the  instru- 
ment, whenever  an  azimuth  determination  is  to  be  made. 
The  second  equation  determines  RQ,  and  through  R0  the 
collimation  corresponding  to  any  position  of  the  microm- 
eter thread  may  be  found ;  e.g.,  let  R  denote  the  reading 
of  the  micrometer  when  the  movable  thread  is  placed 
in  apparent  coincidence  with  any  fixed  thread  of  the 
transit  reticule,  then  will  the  collimation  of  this  thread 
be  given  by  Equation  159.  This  method  of  determining 
collimation  may  be  employed  in  connection  with  time 
determinations,  as  indicated  in  §  77. 

To  apply  these  equations  to  the  reduction  of  a  set 
of  azimuth  observations  we  let  5  represent  the  mean  of 
several  micrometer  readings  made  in  quick  succession 
upon  the  star,  and  similarly  we  represent  by  M  the  mean 
of  several  readings  to  the  mark.  Introducing  these 
quantities  into  Equation  159,  we  find  for  the  star  and 
mark,  respectively, 

c=±k(S-RQ),     c'  =  ±k(M-R0),          (162) 


192  FIELD  ASTRONOMY. 

and  substituting  these  values  in  Equation  158,  we  obtain 
A-A'  =  ±k\(S-M)+  o-  (S-R0)}+btanh.  (163) 
This  equation  may  be  used  for  the  reduction  of  the  ob- 
servations; but  if  the  instrument  has  been  frequently 
reversed  during  the  progress  of  the  work,  it  will  be  more 
convenient  to  combine  in  one  computation  consecutive 
observations  in  its  positive  and  negative  positions. 
Employing  the  subscripts  i  and  2  to  distinguish  obser- 
vations made  in  these  respective  positions,  we  obtain,  by 
taking  the  mean  of  the  resulting  equations,  Circle  W. 
and  Circle  E.,  and  introducing  a  correction  for  diurnal 
aberration, 


-  b  tan  h  +  Di.  Ab.     (i  64) 

In  this  equation  A  represents  the  mean  of  the  azi- 
muths of  the  star  at  the  several  times  of  observation,  and 
for  this  mean  there  may  usually  be  substituted  the  azimuth 
corresponding  to  the  mean  of  the  times  (see  §  68).  The 
level  constant,  6,  represents  the  mean  of  the  inclinations 
of  the  horizontal  axis  in  the  two  positions  of  the  instru- 
ment, and  it  should  be  noted  that  if  the  level  is  left  undis- 
turbed upon  the  axis  during  the  reversal,  the  resulting 
bubble  readings,  Circle  W.  and  Circle  E.,  will  give  this 
mean  inclination,  free  from  the  effect  of  inequality  of 
pivots.  In  the  case  of  a  hanging  level  all  necessity  for 
lifting  it  from  the  axis  or  in  any  way  disturbing  its  rela- 
tion to  the  instrument  is  thus  removed. 

85.  Diurnal  Aberration.  —  In  explanation  of  the  last 
term  of  Equation  164  we  note  that  the  precision  attain- 


THE  TRANSIT  INSTRUMENT.  193 

able  with  a  transit  instrument  is  sufficient  to  demand  a 
consideration  of  the  effect  of  diurnal  aberration,  and  the 
student  may  show  from  the  data  in  §  27  that  for  any 
star  near  the  pole  this  effect  is  fully  compensated  by 
adding  to  the  computed  azimuth  of  the  mark  the  cor- 
rection, 

Di.  Ab.  =  +o".32  cos  0sec  h.  (165) 

Since  0  and  h  are  very  nearly  equal  for  close  circumpolar 
stars,  this  correction  is  practically  constant  and  equal 
to  +o".32. 

86.  Example. — Azimuth  Determination  with  Transit. — 
The  following  example  represents  the  record  and  reduc- 
tion of  a  single  set  of  azimuth  observations  made  with 
the  large  transit  instrument  of  the  ' '  broken ' '  type 
shown  in  the  Frontispiece.  Note  that  the  recorded 
sidereal  times  show  that  the  observations  were  made  at 
about  9  or  10  o'clock  A.M.,  2ih  or  22h  astronomical 
reckoning,  i.e.,  in  broad  daylight.  Values  of  the  instru- 
mental constants  and  other  data  required  for  the  reduc- 
tion follow  immediately  after  the  record,  the  value  of 
the  chronometer  correction,  AT,  having  been  determined 
for  this  purpose  from  time  observations  immediately 
following  the  azimuth  work.  At  the  time  of  the  azimuth 
observations  Polaris  was  near  upper  culmination,  and  an 
inspection  of  the  micrometer  readings  to  the  star,  shows 
that  they  diminish  progressively  for  Circle  W.,  which  is 
therefore  the  positive  position  of  the  instrument  and  is 
to  receive  the  subscript  i  in  the  reductions.  The  azimuth 
i.f  Polaris  is  computed  from  Equation  124. 


194 


FIELD  ASTRONOMY. 


POLARIS   AND    AZIMUTH    MARK. 

Wednesday,  May  7,  1902. 
Bamberg  Transit.     Chronometer,  S.     Observer,  C. 


Circle. 

Azimuth  Mark. 

Polaris. 

Remarks. 
Levels. 

Micrometer. 

Chronometer. 

Micrometer. 

Chronometer. 

E. 
W. 

l5-637 
.622 
.601 
.606     ' 
.610 

15.083 
.067 
.061 
.064 
.087 

h.      m. 
i     8 

i     20 

9.803 
10.019 
.189 

19.799 
•578 
•394 

h.  m.      s. 
I    12    47 
13    26 
13    56 

i    16   18 
16  59-5 
17   35 

Mark  very 
unsteady. 

W.       E. 
29.4    59.2 
60.4   30.8 

+  i-3 
/*=44°i8/ 

Instrumental  Constants. 


REDUCTION. 


M2 

15  .615 

* 

43°    4'37"-o 

10.004 

d 

88    47        1-7 

5i 

19.590 

h.    m.     s. 

MI 

i5-°7 

a 

1    23       5.4 

T  +  AT 

I    15    41.2 

log  (5^50 

0.9816?* 

t 

23  S2  35-8 

log  o 

9.5990 

t 

358     8  57.0 

+  (52-M2) 

—  5.611 

cos  t 

9-999774 

-(5,  -M,) 

—4.518 

tan  0 

9.970825 

+  <r(S,-SJ 

-3.807 

cot  d 

8.326946 

Sum 

-I3-936 

sec  <f> 

0.136417 

sin  t 

8.509169?* 

log  Sum 

i  .14414?* 

p  \  B\ 

0.008532 

log  £& 

I-459I7 

(  B2 

171 

—tan  A 

6.981235?* 

cos  <j> 

9.863 

sec  /& 

0.147 

log* 

8-297545 

0.32 
Di.  Ab. 

9-5°5 
9-5I5 

log*2 

6-5951 

A 

180°    3/i7/'-54 

Micrometer 

-6  41    -15 

Level 

—  o     o  .65 

Di.  Aberration 

+  o     o  .33 

Azimuth 

J79    S6  36  .07 

THE  TRANSIT  INSTRUMENT. 

To  eliminate  any  error  that  might  exist  in  the  assumed 
value  of  a  revolution  of  the  micrometer  screw,  a  second 
set  of  comparisons  of  the  star  and  mark  was  made  a  half- 
hour  later  than  those  reduced  above,  when  the  star  was 
on  the  opposite  side  of  the  mark  and  at  an  approximately 
equal  distance  from  it.  The  resulting  value  of  the  azi- 
muth of  the  mark  was  A'  =  179°  56'  36". 20. 

When  the  highest  accuracy  is  required  a  considerable 
number  of  such  sets  of  observations  should  be  made, 
extending  over  at  least  three  or  four  days  and,  when  pos- 
sible, so  timed  that  the  star  will  be  observed  at  opposite 
points  of  its  diurnal  path,  i.e.,  near  its  upper  and  lower 
culmination,  in  order  to  eliminate  errors  in  its  assumed 
right  ascension  and  declination.  A  study  of  the  errors 
of  the  micrometer  screw  should  also  be  made  (see  §  72)1, 
and  the  resulting  corrections  for  periodic  and  progressive 
error  applied  to  the  several  readings.  The  azimuth  of 
the  star  should,  in  general,  be  computed  with  six-place 
logarithmic  tables,  but  when,  as  in  this  case,  the  star  is 
very  near  the  meridian  five  places  of  decimals  are  quite 
sufficient. 

Query. — Is  it  legitimate  in  this  case  to  neglect  the 
corrections,  AA0,  represented  by  Equation  127? 

For  an  extended  treatise  showing  the  methods  used 
in  the  U.  S.  Coast  and  Geodetic  Survey  for  the  deter- 
mination of  time  and  azimuth  with  a  transit  instrument, 
reference  may  be  made  to  Appendix  7,  Annual  Report  of 
the  Survey  for  1897-98. 


REFERENCE  WORKS. 

FOR  a  more  detailed  treatment  of  the  problems  of 
spherical  and  practical  astronomy  than  is  contained  in 
the  preceding  pages,  the  advanced  student  may  consult 
with  profit  the  following  works : 

1.  Chauvenet.     A  Manual  of  Spherical  and  Practical  Astronomy. 
2  vols.     Philadelphia.     Various  editions. 

2.  Hayford.     Determination  of  Time,  Longitude,  Latitude,  and 
Azimuth.      Appendix  No.  7,  U.  S.  Coast  and  Geodetic  Survey.     Sixty- 
seventh  Annual  Report.     Washington.     1899. 

3.  Albrecht.     Formeln  und  Hiilfstafeln  fiir  Geographische  Orts- 
bestimmungen.     Leipzig.     Third  Edition.     1894. 

4.  Albrecht.     Anleitung  zum  Gebrauche  des  Zenitteleskops  auf 
den  Internationalen  Breitenstationen.     Berlin.     1902. 

5.  Bamberg.     Anweisung  zur  Behandlung  der  Universal  Instru- 
mente  und  Theodoliten  mit  mikroskopischer  Ablesung,  etc.     Berlin. 
1883. 

Of  the  above  works  No.  i  is  the  standard  treatise  upon  the  subject; 
an  elaborate  manual  known  and  used  among  astronomers  of  every 
land.  No.  2  is  much  more  limited  in  its  scope,  but  presents  well  the 
methods  in  use  in  the  U.  S.  Coast  and  Geodetic  Survey.  No.  3  pre- 
sents similarly  the  current  German  practice  and  is  accompanied  by  a 
valuable  series  of  numerical  tables.  No.  4  is  a  special  monograph, 
and  No.  5  a  trade  pamphlet  presenting  details  of  the  use  and  care  of 
geodetic  instruments  not  readily  accessible  elsewhere. 

196 


TABLES. 


197 


TABLES  FOR  THE  DETERMINATION  OF 

AZIMUTH,   LATITUDE,   AND  TIME 

WITHOUT  THE  USE  OF  AN   ALMANAC. 

See  Sections  32-33. 
TABLE  I. 


/ 

a 

3 

t 

0* 

-o'  + 

+  74'  + 

24h 

25 

2 

I 

-25  + 

+  72  + 

23 

24 

7 

2 

-49  + 

+  65  + 

22 

20 

12 

3 

-69  + 

+  53  + 

21 

15 

16 

4 

-84  + 

+  37  + 

20 

9 

»7 

5 

-93  + 

+  20  + 

19 

3 

19 

6 

-96  + 

+     1  + 

18 

4 

*9 

7 

-92  + 

-18- 

J7 

10 

18 

8 

-82  + 

-36- 

16 

15 

15 

9 

-67  + 

-51- 

15 

2O 

12 

10 

-47  + 

-63- 

14 

23 

7 

ii 

-24  + 

-70- 

13 

24 

2 

12 

-    0  + 

-72- 

12 

*=Local  Mean  Time +4™ \D-E\  (i  - 


198 


FIELD  ASTRONOMY. 


TABLE  II. 


<£ 

1900 

I9IO 

1920 

1930 

20° 

0.82 

0.78 

0-75 

0.71 

7 

7 

6 

6 

3° 

0.89 

0.85 

0.81 

0.77 

ii 

it 

II 

ii 

40 

I  .OO 

0.96 

0.92 

0.88 

19 

18 

17 

16 

5° 

I.I9 

I.I4 

1  .09 

I  .04 

See  pp.  77,  78. 

TABLE  III. 
F2  and  a. 


TABLE  IV. 
E. 


Year. 

F* 

a 

1900 
1910 

I  .00 

0.96 

lh  23m 
I   27 

1920 
1930 

o  .92 
0.88 

I   32 

i  37 

1900 

April  12 

•5 

1910 

April  14 

.0 

1920 

April  14 

5 

1901 

12 

.8 

1911 

14 

3 

1921 

14 

8 

1902 

13 

.2 

1912 

13 

7 

1922 

15 

2 

1903 

13 

•5 

1913 

14 

o 

1923 

15 

5 

1904 

12 

•9 

1914 

14 

4 

1924 

14 

9 

1905 

13 

.  2 

I9I5 

14 

7 

1925 

15 

2 

1906 

X3 

.6 

1916 

14 

i 

1926 

6 

1907 

13 

•9 

1917 

14 

4 

1927 

15 

9 

1908 

13 

•3 

1918 

14 

8 

1928 

15 

1909 

13 

.6 

1919 

15 

i 

1929 

15 

6 

1910 

14 

.0 

1920 

14 

5 

1930 

16 

0 

TABLES. 


199 


TABLE   V.     TIME  STARS. 


Star. 

Mag. 

a  1900. 

Ann. 
Var. 

5 

Sh  P.M. 

i  Ceti  

3.3 

h      m      s 

o   14  24 

8 
+  3.1 

o        f 
—     O     22 

Nov     25 

fl  Ceti              

2  .  2 

O    "?8    37 

3    O 

—  l8     32 

IT'ec         i 

£  Ceti           

3-8 

I    IQ       5 

3  .  0 

—     8    42 

Dec       ii 

cc  Piscium   

7  .  Q 

i    56    56 

3    I 

4-217 

Dec      2  1 

y  Ceti          , 

*.6 

2     38     IO 

3    I 

+     2    4O 

Dec       3i 

1  8 

3   28   14 

o   8 

—   o  48 

Tan       1  2 

f  Eridani   

•2  ,  "I 

3     5323 

2     8 

—  13    48 

Tan       10 

v  Eridani  

4  •  * 

4   31    20 

3    O 

—     333 

jctai.        J.y 

B  Orionis    

o  .  3 

50    45 

2    O 

—   8   19 

jaii.      29 
Feb        7 

2  .  2 

543       2 

2    8 

—     O    4.2 

Feb      1  6 

^Canis  Maj  

2  .0 

—  14. 

b    18    19 
6  4.0  4.6 

2.6 

2    6 

-J7   54 
—  16   35 

Feb.     25 

TJ  Canis  Maj  

2    4 

7   20   10 

2    4. 

iw     OO 

—  29      6 

o  Ar£nis 

2    O 

8      3   18 

2    6 

—  24.        I 

March  2  3 

<i  Mali           

3   6 

8   30   36 

2    4 

—  32     5O 

April      2 

d  Jlydra?  .     

2     2 

92  2    4.2 

2     O 

O^     Ov 

—   8   14 

Anril    1  3 

A  Hydra?      

3O 

IO        5    4.4. 

2     O 

—  II     52 

Anril    2  3 

2  .  7 

10  44  43 

•  •  y 
3O 

±  j.     ^.tf 

—  15    4.O 

Mav        3 

d  Crateris  

3-8 

II     14    22 

3    O 

—  14,    14. 

May     1  1 

3-8 

ii   45    31 

31 

+     2     2O 

May     19 

2  .  7 

12     IO    41 

31 

—  17       O 

May     25 

•3    o 

12     36     37 

3O 

—    O    54. 

Tune       i 

Spica       

I     2 

13     10     57 

3    2 

—  IO    3O 

Tune     1  2 

ic  Hydra?           

5    e 

14.       O    42 

34 

—  26    12 

une    22 

3  .  o 

14    37   40 

3  .2 

—    5    14 

July       i 

/?  Libra?   

2.8 

15    II   40 

3  .  2 

—    0       I 

July     10 

2  .  7 

ie    en    30 

2  .  C 

—  10    32 

Tuly     2  3 

Antares         

I  .  3 

16    23    IQ 

3  •  7 

—  26   13 

July     29 

2.6 

17      4   41 

3   -4 

—  15    36 

Aug.       8 

3  •  7 

17  31   55 

3  .4 

—  15    2O 

Aug.     14 

3  .  o 

18   16   ii 

3  •  z 

—   2  55 

Aug.     26 

2  .  I 

18  40     7 

3-7 

—  26  25 

Sept.      3 

p  Sagittarii      

•5      O 

IQ    15     55 

3-5 

-18     3 

Sept.    10 

TJ  Aquilae          

4O 

IO    47    26 

3-  x 

+   o  45 

Sept.    1  8 

3  .  2 

20    15    26 

3.4 

—  J5     5 

Sept.    25 

£  Aouarii 

3n 

2O    4.2     10 

3  .  2 

—   9  52 

Oct.        2 

7  .  3 

21     26    21 

3.2 

-60 

Oct.      13 

5  .  2 

22        O    42 

7.  I 

—  o  48 

Oct.      21 

r  Aciuarii            .    , 

3   8 

22     23     44 

3  .  I 

—   o  32 

Oct.      28 

Fomalhaut  .... 

i    3 

22     52     I  I 

3    3 

—  30     9 

Nov.      4 

98  Aquarii     

4T 

23     17    4.6 

3    2 

—  20  39 

Nov.     10 

d  Sculp  toris  

4    6 

23    4.3    46 

4-31 

—  28  41 

Nov.     i  (j 

INDEX. 


Aberration,  diurnal.  57,  192 
Accurate  determinations,  defined,  6 1 

General  principles,  141 
Addition  logarithms.  20,  34 
Adjustments,  of  level,  103 

Of  theodolite,  117,  120 

Of  sextant,  133 

Of  transit,  168,  191 
Almanac,  The,  46 
Altitude,  26 

Reduction  of,  56 
American  Ephemeris,  46 
Angles,  computation  of,  16 
Apparent  solar  time,  35,  39 
Approximate  determinations,  6l 
Approximate  formulae,  9 

Numerical  limits  for,  IO 
Artificial  horizon,  136 
Astronomical  triangle,  31 
Azimuth,  defined,  26 

Computation  of,  65,  152 
Azimuth  determination,  from  sun,  28, 
63,  67 

From  Polaris,  70 

From  star  at  elongation,  85,  89 

From  two  stars,  90,  96 

With  theodolite,  149,  156 

With  transit,  187,  193 
Azimuth  star,  180 

Barometer,  reduction  of,  52 
Bibliography,  196 

Celestial  sphere,  22 
Chronograph,  181 
Chronometer,  care  of,  138 

Correction,  44 

Rate.  45 

Beat,  138 

Precepts  for  use  of,  139 

Comparisons,  139  < 

Circle  readings,  errors  of,  125 
Circummeridian  altitudes,  79 

Graphical  treatment  of,  80 


Circumpolar  stars,  47 
Clock  stars,  180 
Colatitude,  29 
Collimation,  120 

Elimination  of,  12  1 

Factor,  176 

Mechanical  -determination  of,  191 
Coordinates,  systems  of,  24 

Their  uses,  27 

Mutual  relations,  28 

Transformation  of,  30 
Crelle,  multiplication  table,  21 


35 

'Declination,  26 
Determinations  of  azimuth,  latitude, 

time,  60 

Diurnal  aberration,  57,  192 
Dip  of  horizon,  49 

Elongation,  defined,  86 

Formulae  for,  86 

Azimuth  determination  at,  88 

Limits  for  polar  star,  88 
Engineer's  transit,  in 
Equator,  celestial,  23,  26 
Equation  of  time,  40 
Eye  and  ear  observing,  181 

Precision  of,  182 
Equinox,  vernal,  24 

Gradienter,  158 
Calibration  of,  161 
Value  of  a  revolution,  165,  167 

Horizon,  defined,  23- 

Dip  of,  49 
Hour  angle,  26 

Of  Polaris,  71 


Index  correction,  131,  135 
Inequality  of  pivots,  no 


201 


202 


INDEX. 


Latitude,  29 

By  meridian  altitude,  6 1 

By  circummeridian  altitudes,  79 

By  zenith  telescope,  155,  167 
Least  Squares,  178 
Level,  spirit,  99 

Errors  of,  93 

Corrections,  115,  152,  159 
Logarithmic  computation,  5,  12,  1 6 

Accuracy  of,  18 

Tables,  20 
Longitude  and  time,  37 

Magnitudes,  stellar,  47 
Mayer's  equation,  173 
Mean  solar  time,  35,  39 
Meridian,  23,  26 
Micrometer,  calibration  of,  161 

Nadir,  22 

Negative  sign  for  logarithms,  4 
Noon,  36 

Numerical  solution  of  triangle,  5 
Computations,  12,  16 

Observing  list,  142,  147 
Orientation,  60,  70 

Tables  for  Polaris,  78,  197 

Theory  of  tables,  76 

Parallactic  angle,  32 
Parallax,  53 

Personal  equation,  182,  183 
Pivots,  inequality  of,  1 10 
Polaris,  orientation  by,  70,  73 
Poles,  celestial,  22,  26 
Prime  vertical,  23 

Radians,  9,  II 
Reduction  to  meridian, 

With  given  hour  angle,  8l 

With  given  azimuth,  83 

For  zenith  telescope,  161 
Refraction,  nature  of,  50 

Formulae  for,  51 

Coefficients  for  zenith  telescope,  160 
Repetitions,  method  of,  126 

Influence  of  axis  error,  128 
Repsold  Transit  Micrometer,  184. 
Reversal  of  instrument,  1 13 

Effect  of,  122,  184 
Right  ascension,  26 
Rough  determinations,  defined,  60 

Schedule  for  computation,  13 
Semi-diameter,  52 


Sextant,  129 

Adjustments  of,  132,  133 

Eccentricity  of,  135 

Precepts  for  use  of,  137 
Sidereal  time,  30,  38 
>  Conversion  of,  40,  43 

Mean  noon,  42 
Sidereal  chronometer,  45 
Solar  time,  35,  38 

Conversion  of,  40,  43 
Spherical  trigonometry,  I 

Fundamental  formulae  of,  4 

Derived  formulae,  8 

Right-angled  triangles,  9 
Spirit-level,  99 

Value  of  a  division,  100,  105 

Theory  of,  101 

Adjustment  of,  103 

Precepts  for  use  of,  104 
Stars,  coordinates  of,  47 

Visibility  in  telescope,  48,  69 
Subtraction  logarithms,  34 
Sub  polo,  166,  174 

Sun's   altitude,  refraction   correction,. 
62 

Observed  by  projection,  64 

Theodolite,  in 

Theory  of,  117 

Errors  of  adjustment,  117,  I2O 

Precepts  for  use  of,  128 
Time,  different  systems,  35 

Determination  of,  60 

From  altitudes,  63,  84 

Meridian  transits,  68,  74 

Two  stars,  90,  96 

Subsidiary  determination  of,  98 

Equal  altitudes,  142,  146 

Transit  instrument,  177,  184 

Precision  of  determination,  182 
Transit  factors,  173 

Signs  of,  175 
Transit  instrument,  168 

Adjustment  of,  169 

Theory  of,  170 
Trigonometric  functions,  15 

Vernal  equinox,  24,  26 
Vertical,  22 

Plane,  23 

Circle,  23 

Coordinate,  24 

Zenith,  22,  26 

Zenith  distance,  determination  of,  1 I*. 

Zenith  telescope,  157 


-  o8rRTHA^ 

.  .:        "••• 


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Principles  of  Animal  Nutrition 8vo,  4  oo 

Budd  and  Hanson's  American  Horticultural  Manual: 

Part  I. — Propagation,  Culture,  and  Improvement 12 mo,  i  50 

Part  II. — Systematic  Pomology 12010,  i  50 

Downing's  Fruits  and  Fruit-trees  of  America 8vo,  5  06 

Elliott's  Engineering  for  Land  Drainage i2mo,  i  50 

Practical  Farm  Drainage 1 2mo,  i  oo 

Green's  Principles  of  American  Forestry.     (Shortly.) 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (Woll.) I2mo,  2  oo 

Kemp's  Landscape  Gardening i2mo,  2  50 

Maynard's  Landscape  Gardening  as  Applied  to  Home  Decoration i2mo,  i  50 

Sanderson's  Insects  Injurious  to  Staple  Crops 121110,  i  50 

Insects  Injurious  to  Garden  Crops.     (In  preparation.) 
Insects  Injuring  Fruits.     (In  preparation.) 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

Woll's  Handbook  for  Farmers  and  Dairymen i6mo,  i  50 

ARCHITECTURE. 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Berg's  Buildings>nd  Structures  of  American  Railroads 4to,  5  oo 

Birkmire's  Planning  and  Construction  of  American  Theatres 8vo,  3  oo 

Architectural  Iron  and  Steel 8vo,  3  50 

Compound  Riveted  Girders  as  Applied  in  Buildings 8vo,  2  oo 

Planning  and  Construction  of  High  Office  Buildings 8vo,  3  50 

Skeleton  Construction  in  Buildings 8vo,  3  oo 

Briggs's  Modern  American  School  Buildings 8vo,  4  oo 

Carpenter's  Heating  and  Ventilating  of  Buildings 8vo,  4  oo 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten 8vo,  3^50 

Fireproofing  of  Steel  Buildings 8vo,  2  50 

French  and  Ives's  Stereotomy 8vo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  i  oo 

Theatre  Fires  and  Panics i2mo,  i  50 

1 


Hatneld's  American  House  Carpenter 8vo,    5  oo 

Holly's  Carpenters'  and  Joiners'  Handbook i8mo,        75 

Johnson's  Statics  by  Algebraic  and  Graphic  Methods 8vo,    2  oo 

Kidder's  Architect's  and  Builder's  Pocket-book i6mo,  morocco,  4  oo 

Merrill's  Stones  for  Building  and  Decoration 8vo,    5  oo 

Monckton's  Stair-building 4to,    4  oo 

Patton's  Practical  Treatise  on  Foundations 8vo,    5  oo 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry 8vo,    i   50 

Snow's  Principal  Species  of  Wood 8vo,    3  50 

Sondericker's  Graphic  Statics  with  Applications  to  Trusses,  Beams,  and  Arches. 
(Shortly.) 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,    6  oo 

Sheep,    6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture   8vo,    5  oo 

Sheep,    5  50 

Law  of  Contracts 8vo,    300 

Woodbury's  Fire  Protection  of  Mills 8vo,    2  50 

Worcester  and  Atkinson's  Small  'Hospitals,  Establishment  and  Maintenance, 
Suggestions  for  Hospital  Architecture,  with  Plans  for  a  Small  Hospital. 

i2mo,    i   25 
The  World's  Columbian  Exposition  of  1893 Large  4to,    i  oo 


ARMY  AND  NAVY. 

Bernadou's  Smokeless  Powder,  Nitro-cellulose,  and  the  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

*  Bruff's  Text-book  Ordnance  and  Gunnery 8vo,  6  oo 

Chase's  Screw  Propellers  and  Marine  Propulsion 8vo,  3  oo 

Craig's  Azimuth 4to,  3  50 

Crehore  and  Squire's  Polarizing  Photo-chronograph 8vo,  3  oo 

Cronkhite's  Gunnery  for  Non-commissioned  Officers 24mo.  morocco,  2  oo 

*  Davis's  Elements  of  Law 8vo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States 8vo,    7  oo 

*  Sheep    7  50 

De  Brack's  Cavalry  Outpost  Duties.     (Carr.) 241110,  morocco,    2  oo 

Dietz's  Soldier's  First  Aid  Handbook i6mo,  morocco,    i  25 

*  Dredge's  Modern  French  Artillery 4to,  half  morocco,  15  oo 

Durand's  Resistance  and  Propulsion  of  Ships 8vo,  5  oo 

*  Dyer's  Handbook  of  Light  Artillery i2mo,  3  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

*  Fiebeger's  Text-book  on  Field  Fortification Small  8vo,  2  oo 

Hamilton's  The  Gunner's  Catechism i8mo,  i  oo 

*  Hoff's  Elementary  Naval  Tactics 8vo,  i  50 

Ingalls's  Handbook  of  Problems  in  Direct  Fire 8vo,  4  oo 

*  Ballistic  Tables 8vo,    i  50 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.   Vols.  I.  and  II.  .8vo,each,  6  oo 

*  Mahan's  Permanent  Fortifications.     (Mercur.) 8vo,  half  morocco,  7  50 

Manual  for  Courts-martial i6mo   morocco,  i   50 

*  Mercur's  Attack  of  Fortified  Places i2mo,  2  oo 

*  Elements  of  the  Art  of  War 8vo,    4  oo 

Metcalf 's  Cost  of  Manufactures — And  the  Administration  of  Workshops,  Public 

and  Private -.  . . .  t '. 8vo,    5  oo 

*  Ordnance  and  Gunnery .....". i2mo,    5  oo 

Murray's  Infantry  Drill  Regulations i8mo,  paper,        10 

*  Phelps's  Practical  Marine  Surveying 8vo,    2  50 

Powell's  Army  Officer's  Examiner I2mo,    4  oo 

Sharpe's  Art  of  Subsisting  Armies  in  War i8mo,  morocco,    i   50 

2 


*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

*  Wheeler's  Siege  Operations  and  Military  Mining 8vo,  2  oo 

Winthrop's  Abridgment  of  Military  Law i2mo,  2  50 

Woodhull's  Notes  on  Military  Hygiene i6mo,  i   50 

Young's  Simple  Elements  of  Navigation i6mo,  morocco,  i  oo 

Second  Edition,  Enlarged  and  Revised i6mo,  morocco,  2  co 


ASSAYING. 

Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

i2mo,  morocco,!  J  SO 

Furman's  Manual  of  Practical  Assaying 8vo,  3  oo 

Miller's  Manual  of  Assaying I2mo,  i  oo 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process i2mo,  i  so 

03»      S 

ASTRONOMY. 

Comstock's  Field  Astronomy  for  Engineers v» . ^  : ,; . . .  .  8vo,  2  50 

Craig's  Azimuth 4to,  3  50 

Doolittle's  Treatise  on  Practical  Astronomy 8vo,  4  oo 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  co 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  50 

*  Michie  and  Harlow's  Practical  Astronomy 8vo,  3  oo 

*  White's  Elements  of  Theoretical  and  Descriptive  Astronomy 12010,  2  oo 

BOTANY. 

Davenport's  Statistical  Methods,  with  Special  Reference  to  Biological  Variation. 

i6mo,  morocco,  i   25 

Thome  and  Bennett's  Structural  and  Physiological  Botany i6mo,  2  25 

Westermaier's  Compendium  of  General  Botany.     (Schneider.) 8vo,  2  oo 


CHEMISTRY. 

Adriance's  Laboratory  Calculations  and  Specific  Gravity  Tables i2mo,  i  25 

Allen's  Tables  for  Iron  Analysis 8vo,  3  oo 

Arnold's  Compendium  of  Chemistry.     (Mandel.)     (In  preparation.) 

Austen's  Notes  for  Chemical  Students i2mo,  i  50 

Bernadou's  Smokeless  Powder. — Nitro-cellulose,  and  Theory  of  the  Cellulose 

Molecule '. 12010,  2  50 

Bolton's  Quantitative  Analysis 8vo,  i  50 

*  Browning's  Introduction  to  the  Rarer  Elements 8vo,  i  50 

Brush  and  Penfield's  Manual  of  Determinative  Mineralogy 8vo,  4  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.  (Boltwood.)  .  .  .  .8vo,  3  oo 

Conn's  Indicators  and  Test-papers i2mo,  2  oo 

Tests  and  Reagents 8vo,  3  oo 

Copeland's  Manual  of  Bacteriology.     (In  preparation.) 

Craft's  Short  Course  in  Qualitative  Chemical  Analysis.  (Schaeffer.). .  .  .  i2mo,  2  oo 

Drechsel's  Chemical  Reactions.     (Merrill.) i2mo,  i  25 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.)     (Shortly.) 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

3 


Effront's  Enzymes  and  their  ^Applications.     (Prescott.) 8vo,  3  oo 

Erdmann's  Introduction  to  Chemical  Preparations.     (Dunlap.) xarno,  i  25 

Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

i2mo,  morocco,  i  50 

Fowler's  Sewage  Works  Analyses I2mo,  2  oo 

Fresenius's  Manual  of  Qualitative  Chemical  Analysis.     (Wells.) 8vo,  5  oo 

Manual  of  Qualitative  Chemical  Analysis.     Parti.    Descriptive.     (Wells.) 

8vo,  3  oo 

System   of   Instruction   in    Quantitative    Chemical   Analysis.      (Cohn.) 
2  vols.     (Shortly.} 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  oo 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i  25 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (Wo  11.) i2mo,  2  oo 

Hammarsten's  Text-book  of  Physiological  Chemistry.     (Mandel.) 8vo,  4  oo 

Helm's  Principles  of  Mathematical  Chemistry.     (Morgan.) i2mo.  i  50 

Hinds's  Inorganic  Chemistry 8vor  3  oo 

*       Laboratory  Manual  for  Students I2mo,        75 

Holleman's  Text-book  of  Inorganic  Chemistry.     (Cooper.) 8vo,  2  50 

Text-book  of  Organic  Chemistry.     (Walker  and  Mott.) 8vo,  2  50 

Hopkins's  Oil-chemists'  Handbook 8vo,  3  oo 

Keep's  Cast  Iron 8vo ,  250 

Ladd's  Manual  of  Quantitative  Chemical  Analysis 12 mo.  i  oo 

Landauer's  Spectrum  Analysis.    (Tingle.) ' 8vo,  3  oo 

Lassar-Cohn's  Practical  Urinary  Analysis.     (Lorenz.) i2mo,  i  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control.     (In  preparation.) 

Lob's  Electrolysis  and  Electrosynthesis  of  Organic  Compounds.  (Lorenz.)  i2mo,  i  oo 

Mandel's  Handbook  for  Bio-chemical  Laboratory i2mo,  i  50 

Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

3d  Edition,  Rewritten 8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  125 

Meyer's  Determination  of  Radicles  in  Carbon  Compounds.     (Tingle.).  .  i2mo,  i  oo 

Miller's  Manual  of  Assaying i2mo,  i  oo 

Mixter's  Elementary  Text-book  of  Chemistry i2mo,  i  50 

Morgan's  Outline  of  Theory  of  Solution  and  its  Results i2mo,  i  oo 

Elements  of  Physical  Chemistry i2mo,  2  oo 

Nichols's  Water-supply.     (Considered  mainly  from  a  Chemical  and  Sanitary 

Standpoint,  1883.) 8vo,  2  50 

O'Brine's  Laboratory  Guide  in  Chemical  Analysis 8vo,  2  oo 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Ost  and  Kolbeck's  Text-book  of  Chemical  Technology.     (Lorenz — Bozart.) 
( In  preparati on.) 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,        50 

Pictet's   The   Alkaloids   and   their   Chemical  Constitution.      (Biddle.)      (In 
preparation. ) 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) i2mo,  i  50 

Poole's  Calorific  Power  of  Fuels 8vo.  3  oo 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Richards  and  Woodman's  Air,Water,  and  Food  from  a  Sanitary  Standpoint .  8vo,  2  oo 

Richards's  Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  i  oo 

Cost  of  Food,  a  Study  in  Dietaries i2mo,  i  oo 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  i   50 

Ricketts  and  Russell's  Skeleton  Notes  upon  Inorganic  Chemistry.     (Part  I. — 

Non-metallic  Elements.) 8vo,  morocco,          75 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  3  50 

4 


Ruddiman's  Incompatibilities  in  Prescriptions 8vo,  2  oo 

Schimpf's  Text-book  of  Volumetric  Analysis i2mo,  2  50 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  oo 

Handbook  for  Sugar  Manufacturers  and  their  Chemists.  .  i6mo,  morocco,  2  oo 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat 8vo,  i  50 

*  Descriptive  General  Chemistry 8vo  3  oo 

TreadwelTs  Qualitative  Analysis.     (Hall.) .  . 8vo,  3  oo 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Van  Deventer's  Physical  Chemistry  for  Beginners.     (Boltwood.) i2tno,  i  50 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Wells's  Laboratory  Guide  in  Qualitative  Chemical  Analysis 8vo,  i  50 

Short  Course  in  Inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students i2mo,  i  50 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Wiechmann's  Sugar  Analysis Small  8vo.  2  50 

Wilson's  Cyanide  Processes I2mo,  i  50 

Chlorination  Process i  amo,  i  50 

Wulling's  Elementary  Course  in  Inorganic  Pharmaceutical  and  Medical  Chem- 
istry  i2mo,  2  oo 


CIVIL   ENGINEERING. 

BRIDGES  AND    ROOFS.       HYDRAULICS.      MATERIALS    OF   ENGINEERING. 
RAILWAY  ENGINEERING. 

Baker's  Engineers'  Surveying  Instruments I2mo,  3  oo 

Bixby's  Graphical  Computing  Table Paper,  ig^X  24!  inches.  25 

**  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Canal.     (Postage, 

27  cents  additional.) 8vo,  net,  3  50 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

Davis's  Elevation  and  Stadia  Tables 8vo,  i  oo 

Elliott's  Engineering  for  Land  Drainage i2mo,  i  50 

Practical  Farm  Drainage i2mo,  i  oo 

Folwell's  Sewerage.     (Designing  and  Maintenance.) 8vo,  3  oo 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten 8vo,    3  50 

French  and  Ives's  Stereotomy 8vo,  2  50 

Goodhue's  Municipal  Improvements i2mo,  i  75 

Goodrich's  Economic  Disposal  of  Towns'  Refuse 8vo,  3  50 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  oo 

Howe's  Retaining  Walls  for  Earth i2mo,  i  25 

Johnson's  Theory  and  Practice  of  Surveying Small  8vo,  4  oo 

Statics  by  Algebraic  and  Graphic  Methods 8vo,  2  oo 

Kiersted's  Sewage  Disposal i2mo,  i  25 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory.)  i2mo,  2  oo 

Mahan's  Treatise  on  Civil  Engineering.     (1873.)     (Wood.) 8vo,  5  oo 

*       Descriptive  Geometry 8vo,  i  50 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  50 

Elements  of  Sanitary  Engineering 8vo,  2  oo 

Merriman  and  Brooks's  Handbook  for  Surveyors i6mo,  morocco,  2  oo 

Nugent's  Plane  Surveying 8vo,  3  50 

Ogden's  Sewer  Design I2mo,  2  oo 

Patton's  Treatise  on  Civil  Engineering 8vo,  half  leather,  7  50 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

Rideal's^Sewage  and  the  Bacterial  Purification  of  Sewage ^. . .  8vo,  3  50 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry *. .  .8vo,  i  50 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) 8vo,  2  50 

5 


-ondericker's   Graphic   Statics,   with   Applications   to   Trusses,   Beams,   and 
Arches.     (Shortly.) 

*  Trautwine's  Civil  Engineer's  Pocket-book i6mo,  morocco,  5  oo 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  5  oo 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  oo 

Warren's  Stereotomy — Problems  in  Stone-cptting 8vo;  2  50 

Webb's  Problems  in  the  Uoe  and  Adjustment  of  Engineering  Instruments. 

i6mo,  morocco,  i  25 

*  Wheeler's  Elementary  Course  of  Civil  Engineering 8vo,  4  oo 

Wilson's  Topographic  Surveying 8vo,  3  50 


BRIDGES  AND  ROOFS. 

Boiler's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges.  .8vo,  2  oo 

*         Thames  River  Bridge 4to,  paper,  5  oo 

Burr's  Course  on  the  Stresses  in  Bridges  and  Roof  Trusses,  Arched  Ribs,  and 

Suspension  Bridges 8vo,    3  50 

Du  Bois's  Mechanics  of  Engineering.     Vol.  II Small  4to,  10  oo 

Foster's  Treatise  on  Wooden  Trestle  Bridges 4to,  5  oo 

Fowler's  Coffer-dam  Process  for  Piers 8vo,  2  50 

Greene's  Roof  Trusses 8vo,  i  25 

Bridge  Trusses 8vo,  2  50 

Arches  in  Wood,  Iron,  and  Stone 8vo,  2  50 

Howe's  Treatise  on  Arches 8vo  4  oo 

Design  of  Simple  Roof-trusses  in  Wood  and  Steel 8vo,  2  oo 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Designing  of 

Modern   Framed   Structures Small  4to,  10  oo 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges: 

Part  I. — Stresses  in  Simple  Trusses , 8vo,  2  50 

Part  II. — Graphic  Statics 8vo,  2  50 

Part  III. — Bridge  Design.     4th  Edition,  Rewritten 8vo,  2  50 

Part  IV. — Higher  Structures 8vo,  2  50 

Morison's  Memphis  Bridge 4to,  10  oo 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers. .  .  i6mo,  morocco,  3  oo 

Specifications  for  Steel  Bridges i2mo,  r  25 

Wood's  Treatise  on  the  Theory  of  the  Construction  of  Bridges  and  Roofs.Svo.  2  oo 
Wright's  Designing  of  Draw-spans: 

Part  I.  — Plate-girder  Draws 8vo,  2  50 

Part  II. — Riveted-truss  and  Pin -connected  Long-span  Draws 8vo,  2  50 

Two  parts  in  one  volume 8vo,  3  50 


HYDRAULICS. 

Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from  an 

Orifice.     (Trautwine.) 8vo,    2  oo 

Bovey's  Treatise  on  Hydraulics 8vo,    5  oo 

Church's  Mechanics  of  Engineering r 8vo,    6  oo 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels paper  ^    i  50 

Coffin's  Graphical  Solution  of  Hydraulic  Problems i6mo,  morocco,    2  50 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,    3  oo 

Folwell's  Water-supply  Engineering 8vo,    4  oo 

Frizell's  Water-power 8vo,    5  oo 

6 


Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration  Works i2mo,  2  50 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Hering  and  Trautwine.) 8vo,  4  oo 

Hazen's  Filtration  of  Public  Water-supply 8vo,  3  oo 

Hazlehurst's  Towers  and  Tanks  for  Water-works 8vo,  2  50 

Herschel's  115  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal| 

Conduits 8vo,  2  oo 

Mason's    Water-supply.     (Considered    Principally   from   a    Sanitary    Stand- 
point.)    3d  Edition,  Rewritten 8vo,  4  oo 

Merriman's  Treatise  on  Hydraulics,     gth  Edition,  Rewritten 8vo,  5  oo 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  oo 

Schuyler's   Reservoirs   for  Irrigation,  Water-power,   and  Domestic   Water- 
supply Large  8vo,  5  oo 

**  Thomas  and  Watt's  Improvement  of  Riyers.     (Post.,  44  c.  additional),  4to,  6  oo 

Turneaure  and  Russell's  Public  Water-supplies 8vo.  5  oo 

Wegmann's  Desien  and  Construction  of  Dams 4to,  5  oo 

Water-suoolv  of  the  City  of  New  York  from  1658  to  1895 4to,  10  oo 

Weisbach's  Hydraulics  and  Hydraulic  Motors.     (Du  Bois.) 8vo,  5  oo 

Wilson's  Manual  of  Irrigation  Engineering Small  8vo,  4  oo 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50 

Elements  of  Analytical  Mechanics 8vo,  3  oo 

''  to:  -tsJdc  '  •:''. 

MATERIALS  OF  'ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction 8vo,  5  oo 

Roads  and  Pavements 8vo,  5  oo 

Black's  United  States  Public  Works Oblong  4to,  5  oo 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo*  7  So 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.     6th  Edi- 
tion, Rewritten 8vo,  7  SO 

Byrne's  Highway  Construction 8vo,  5  oo 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo,  3  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Du  Bois's  Mechanics  of  Engineering.     Vol.  I Small  4to,  7  50 

Johnson's  Materials  of  Construction Large  8vo,  6  oo 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,   7  So 

Martens's  Handbook  on  Testing  Materials.     (Henning.)     2  vols 8vo,  7  So 

Merrill's  Stones  for  Building- and  Decoration 8vo,  5  oo 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo,  4  oo 

Strength  of  Materials i2mo,  i  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Patton's  Practical  Treatise  on  Foundations 8vo,  5  oo 

Rockwell's  Roads  and  Pavements  in  France i2mo,  i  25 

Smith's  Wire :  Its  Use  and  Manufacture Small  4to,  3  oo 

Materials  of  Machines i2mo,  i  oo 

Snow's  Principal  Species  of  Wood 8vo,  3  50 

Spalding's  Hydraulic  Cement i2mo,  2  oo 

Text-book  on  Roads  and  Pavements i2mo,  2  oo 

Thurston's  Materials  of  Engineering.     3  Parts. .  .^ 8vo,  8  oo 

Part  I. — Non-metallic  Materials  of  Engineering  and  Metallurgy 8vo,  2  oo 

Part  II. — Iron  and  Steel 8vo,  3  50 

Part  III. — A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

7 


Thurston's  Text-book  of  the  Materials  oflConstruction"* 8vo,  5  oo 

Tillson's  Street  Pavements  and  Paving  Materials 8vo,  4  oo 

Waddell's  De  Pontibus.     (A  Pocket-book  for  Bridge  Engineers.) .  .  i6mo,  mor.,  3  oo 

Specifications  for  Steel  Bridges i2mo,  i  25 

Wood's  Treatise  on  the  Resistance  of  Materials,  and>n  Appendix  on  the  Pres- 
ervation of  Timber 8vo,  2  oo 

Elements  of  Analytical  Mechanics 8vo,  3  oo 


RAILWAY  ENGINEERING. 

Andrews's  Handbook  for  Street  Railway  Engineers.     3X5  inches,  morocco,    i  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  oo 

Brooks's  Handbook  of  Street  Railroad  Location i6mo,  morocco,  i  50 

Butts's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  i  50 

Railway  and  Other  Earthwork  Tables 8vo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.    i6mo,  morocco,  4  oo 

Dredge's  History  of  the  Pennsylvania  Railroad:    (1879) Paper,  5  oo 

*  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills,  4to,  half  mor.,  25  oo 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Godwin's  Railroad  Engineers'  Field-book  and  Explorers'  Guide i6mo,  mor.,  2  50 

Howard's  Transition  Curve  Field-book i6mo,  morocco,  i  50 

Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
bankments  8vo,  i  oo 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  i  oo 

Nagle's  Field  Manual  for  Railroad  Engineers i6mo,  morocco,  3  oo 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  oo 

Pratt  and  Alden's  Street-railway  Road-bed 8vo,  2  oo 

Searles's  Field  Engineering i6mo,  morocco,  3  oo 

Railroad  Spiral i6mo,  morocco,  i  50 

Taylor's  Prismoidal  Formulae  and  Earthwork 8vo,  i  50 

*  Trautwine's  Method  of  Calculating  the  Cubic  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams 8vo,  2  oo 

*  The  Field  Practice  of    Laying    Out    Circular    Curves    for    Railroads. 

i2mo,  morocco,  2  50 

*  Cross-section  Sheet Paper,  25 

Webb's  Railroad  Construction.     2d  Edition,  Rewritten ifimo.  morocco,  5  oo 

Wellington's  Economic  Theory  of  the  Location  of  Railways Small  8vo,  5  oo 


DRAWING. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  oo 

Coolidge's  Manual  of  Drawing 8vo,  paper,  i  oo 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective 8vo,  2  oo 

Jones's  Machine  Design: 

Part  I. — Kinematics  of  Machinery 8vo,  i  50 

Part  II. — Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

MacCord's  Elements  of  Descriptive  Geometry 8vo,  3  oo 

Kinematics;   or,  Practical  Mechanism ': 8vo,  5  oo 

Mechanical  Drawing , 4to,  4  oo 

Velocity  Diagrams 8vo,  i  50 

*  Mahan's  Descriptive  Geometry  and  Stone-cutting 8vo,  i  50 

Industrial  Drawing.    (Thompson.) 8vo,  3  50 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

8 


Reid's  Course  in  Mechanical  Drawing  ...............................  8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design.  .8vo,  3  oo 

Robinson's  Principles  of  Mechanism  ................................  8vo,  3  oo 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.)  ..............  8vo,  50 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing  .  .  i2tno,  oo 

Drafting  Instruments  and  Operations  ...........................  i2mo,  25 

Manual  of  Elementary  Projection  Drawing  ----  .................  12  mo,  50 

Manual  of  Elementary  Eroblems  in  the  Linear  Perspective  of  Form  and 

Shadow  .......  ........................................  I2mo,  oo 

Plane  Problems  in  Elementary  Geometry  .......................  i2mo,  25 

Primary  Geometry  .........................................  i2mo,  75 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective  .......  8vo,  3  50 

General  Problems  of  Shades  and  Shadows  ........................  8vo,  3  oo 

Elements  of  Machine  Construction  and  Drawing  ..................  8vo,  7  So 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry  ........  8vo,  2  50 

Weisbach's  Kinematics  and  the  Power  of  Transmission.       (Hermann  and 

Klein.)   ................................  .  ...............  8vo,  5  oo 

Whelpley's  Practical  Instruction  in  the  Art  of  Letter  Engraving  ........  i2mo,  2  oo 

Wilson's  Topographic  Surveying  ..........  '  ..........................  8vo,  3  50 

Free-hand  Perspective  ....................  .  ...................  8vo,  2  50 

Free-hand  Lettering.     (In  preparation.) 

Woolf's  Elementary  Course  in  Descriptive  Geometry  .............  Large  8vo,  3  oo 


'ELECTRICITY  AND   PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.)  ........  Small  8vo,  3  oo 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measurements  .....  i2mo,  i  oo 

Benjamin's  History  of  Electricity  ...................................  8vo,  3  oo 

Voltaic  Cell  ..................................................  8vo,  3  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.    (Boltwood.).  .8vo,  3  oo 

Crehore  and  Squier's  Polarizing  Photo-chronograph  ...................  8vo,  3  oo 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  i6mo,  morocco,  4  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power  .............  i2mo,  3  oo 

Gilbert's  De  Magnete.     (Mottelay.)  .................................  8vo,  2  50 

Holman's  Precision  of  Measurements  ................................  .8vo,  2  oo 

Telescopic  Mirror-scale  Method,  Adjustments,  and  Tests  .....  Large  8vo,  75 

Landauer's  Spectrum  Analysis.    (Tingle.)  ............................  8vo,  3  oo 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard  —  Burgess.)i2mo,  3  oo 

Lob's  Electrolysis  and  Electrosynthesis  of  Organic  Compounds.  (Lorenz.)  i2mo,  i  oo 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.     Vols.I.  and  II.  8vo,'each,  6  oo 

*  Michie.     Elements  of  Wave  Motion  Relating  to  Sound  and  Light  .......  8vo,  4  oo 

Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (FishoacK.  )  ......  i2mo,  2  56 

*  Parshall  and  Hobart's  Electric  Generators  ........  Small  4to.  half  morocco,  10  oo 

*  Rosenberg's  Electrical  Engineering.   (Haldane  Gee  —  Kinzbrunner.).  .  .  .8vo,  i  50 
Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     (In  preparation.) 

Thurston's  Stationary  Steam-engines  ...............................  8vo,    2  50 

*  Tillman's  Elementary  Lessons  in  Heat  ..............................  8vo,    i  50 

Tory  and  1  itcher's  Manual  of  Laboratory  Physics  ..............  Small  8vo,    2  oo 


LAW. 

*  Davis's  Elements  of  Law 8vo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States 8vo,  7  oo 

*  Sheep,  7  50 
Manual  for  Courts-martial i6mo,  morocco,  i  50 

9 


Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture   ' 8vo,  5  oo 

Sheep,  5  So 

Law  of  Contracts 8vo,  3  oo 

Winthrop's  Abridgment  of  Military  Law i2mo,  2  50 


MANUFACTURES. 

Bernadou's  Smokeless  Powder — Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

Holland's  Iron  Founder i2mo,  2  50 

"  The  Iron  Founder,"  Supplement i2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the 

Practice  of  Moulding i2mo,  3  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Eff rent's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  oo 

Fitzgerald's  Boston  Machinist i8mo,  i  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  oo 

Hopkins's  Oil-chemists'  Handbook 8vo,  3  oo 

Keep's  Cast  Iron 8vo,  2  50 

Leach's  The  Inspection  and  Analysis  of jFood  with  Special  Reference  to  State 

Control.     (In  preparation.} 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Metcalfe's  Cost  of  Manufactures — And  the  Administration    of  Workshops, 

Public  and  Private 8vo,  5  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Smith's  Press-working  of  Metals 8vo,  3  oo 

Wire :   Its  Use  and  Manufacture Small  4to,  3  oo 

Spalding's  Hydraulic  Cement i2mo,  2  oo 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  oo 

Handbook  for  Sugar  Manufacturers  and  their  Chemists. .  .  i6mo,  morocco,  2  oo 
Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion  8vo,  5  oo 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

West's  American  Foundry  Practice i2mo,  2  50 

Moulder's  Text-book i2mo,  2  50 

Wiechmann's  Sugar  Analysis ^ Small  8vo,  2  50 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Woodbury's  Fire  Protection  of  Mills 8vo,  2  50 


MATHEMATICS. 

Baker's  Elliptic  Functions 8vo,    i  50 

s's  Elements  of  Differential  Calculus i2mo,    4  oo 

oo 
50 
50 
50 
50 
25 
75 
50 


Briggs's  Elements  of  Plane  Analytic  Geometry i2mo, 

Chapman's  Elementary  Course  in  Theory  of  Equations i2mo, 

Compton's  Manual  of  Logarithmic  Computations i2mo, 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo, 

*  Dickson's  College  Algebra Large  i2mo, 

*  Introduction  to  the  Theory  of  Algebraic  Equations    Large  i2mo, 

Halsted's  Elements  of  Geometry 8vo, 

Elementary  Synthetic  Geometry 8vo, 


*  Johnson's  Three-place  Logarithmic  Tables:    Vest-pocket  size paper,  15 

103  copies  for  5  oo 

*  Mounted  on  heavy  cardboard,  8  X  10  inches,  25 

10  copies  for  2  oo 

Elementary  Treatise  on  the  Integral  Calculus Small  8vo,  i  50 

Curve  Tracing  in  Cartesian  Co-ordinates 12010,  i  oo 

Treatise  on  Ordinary  and  Partial  Differential  Equations Small  8vo,  3  50 

Theory  of  Errors  and  the  Method  of  Least  Squares i2mo,  i  50 

*  Theoretical  Mechanics i2mo,  3  oo 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory.)  i2mo,  2  oo 

*  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables , 8vo,  3  oo 

Trigonometry  and  Tables  published  separately Each,  2  oo 

Maurer's  Technical  Mechanics.     (In  preparation.) 

Merriman  and  Woodward's  Higher  Mathematics 8vo,  5  oo 

Merriman's  Method  of  Least  Squares 8vo,  2  oo 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus .  Sm.,  8vo,  3  oo 

Differential  and  Integral  Calculus.     2  vols.  in  one Small  8vo,  2  50 

Wood's  Elements  of  Co-ordinate  Geometry 8vo,  2  oo 

Trigonometry:  Analytical,  Plane,  and  Spherical I2mo,  i  oo 

MECHANICAL   ENGINEERING. 
MATERIALS  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Baldwin's  Steam  Heating  for  Buildings I2mo,  2  50 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  oo 

Benjamin's  Wrinkles  and  Recipes i2mo,  2  oo 

Carpenter's  Experimental  Engineering 8vo,  6  oo 

Heating  and  Ventilating  Buildings 8vo,  4  oo 

Clerk's  Gas  and  Oil  Engine Small  8vo,  4  oo 

Coolidge's  Manual  of  Drawing 8vo,    paper,  i  oo 

Cromwell's  Treatise  on  Toothed  Gearing 12010,  i  50 

Treatise  on  Belts  and  Pulleys 12010,  i  50 

Durley's  Kinematics  of  Machines i 8vo,  4  oo 

Flather's  Dynamometers  and  the  Measurement  of  Power 12 mo,  3  oo 

Rope  Driving i2mo,  2  oo 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i  25 

Hall's  Car  Lubrication : i2mo,  i  oo 

Button's  The  Gas  Engine.     (In  preparation.) 
Jones's  Machine  Design: 

Part   I. — Kinematics  of  Machinery 8vo,  i  50 

Part  II. — Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kent's  Mechanical  Engineer's  Pocket-book i6mo,    morocco,  5  oo 

Kerr's  Power  and  Power  Transmission Svo,  2  oo 

MacCord's  Kinematics;  or,  Practical  Mechanism. 8vo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams. 8vo,  i  50 

Mahan's  Industrial  Drawing.    (Thompson.) 8vo,  3  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design .  .  8vo,  3  oo 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Smith's  Press-working  of  Metals 8vo,  3  oo 

Thurston's  Treatise  on   Friction   and    Lost  Work   in    Machinery  and   Mill 

Work •  -8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics .  i2mo,  i  oo 

11 


Warren's  Elements  of  Machine  Construction  and  Drawing Svo,  7  So 

Weisbach's  Kinematics  and  the  Power  of  Transmission.      Herrmann — 

Klein.) Svo,  5  oo 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .8vo,  5  oo 

Hydraulics  and  Hydraulic  Motors.     (Du  Bois.) 8vo,  5  oo 

Wolff's  Windmill  as  a  Prime  Mover Svo,  3  oo 

Wood's  Turbines 8vo,  2  50 

MATERIALS  OF  ENGINEERING. 

Bovey's  Strength  of  Materials  and  Theory  df  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.     6th  Edition, 

Reset 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Johnson's  Materials  of  Construction Large  8vo,  6  oo 

Keep's  Cast  Iron 8vo.  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) 8vo,  7  50 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo,  4  oo 

Strength  of  Materials i2mo,  i  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo  2  oo 

Smith's  Wire :   Its  Use  and  Manufacture Small  4to,  3  oo 

Materials  of  Machines i2mo,  i  oo 

Thurston's  Materials  of  Engineering 3  vols  ,  Svo,  8  oo 

Part   II. — Iron  and  Steel Svo,  3  50 

Part  III. — A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents Svo,  2  so 

Text-book  of  the  Materials  of  Construction Svo  5  oo 

Wood's  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on  the 

Preservation  of  Timber Svo,  2  oo 

Elements  of  Analytical  Mechanics Svo,  3  oo 

STEAM-ENGINES  AND  BOILERS. 

Carnot's  Reflections  on  the  Motive  Power  of  Heat.     (Thurston.) i2mo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  i6mo,  mor.,  4  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  oo 

Goss's  Locomotive  Sparks Svo,  2  oo 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy i2mo,  2  oo 

Button's  Mechanical  Engineering  of  Power  Plants Svo,  5  oo 

Heat  and  Heat-engines Svo,  5  oo 

Kent's  Steam-boiler  Economy Svo,  4  oo 

Kneass's  Practice  and  Theory  of  the  Injector Svo,  i  50 

MacCord's  Slide-valves Svo,  2  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Peabody's  Manual  of  the  Steam-engine  Indicator i2mo,  i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors Svo,  i  oo 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines Svo,  5  oo 

Valve-gears  for  Steam-engines Svo,  2  50 

Peabody  and  Miller's  Steam-boilers Svo,  4  oo 

Pray's  Twenty  Years  with  the  Indicator Large  Svo,  2  50 

Pupln's  Thermodynamics  of  Reversible,  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) I, i2mo,  i  25 

Reagan's  Locomotives  :  Simple,  Compound,  and  Electric i2mo,  2  50 

Rontgen's  Principles  of  Thermodynamics.     (Du  Bois.) Svo,  5  oo 

Sinclair's  Locomotive  Engine  Running  and  Management 12 mo,  2  oo 

Smart's  Handbook  of  Engineering  Laboratory  Practice i2mo,  2  50 

Snow's  Steam-boiler  Practice Svo,  3  oo 

.12 


Spangler's  Valve-gears 8vo,  2^50 

Notes  on  Thermodynamics lamo,  i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thurston's  Handy  Tables 8vo,  i  50 

Manual  of  the  Steam-engine 2  vols.  8vo,    10  oo 

Part  I.— History,  Structuce,  and  Theory 8vo,  6  oo 

Part  II. — Design,  Construction,  and  Operation 8vo,  6  oo 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo,  5  oo 

Stationary  Steam-engines 8vo,  2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice 12 mo,  i  50 

Manual  of  Steam-boilers,  Their  Designs,  Construction,  and  Operation .  8vo,  5  oo 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) 8vo,  5  oo 

Whitham's  Steam-engine  Design 8vo,  5  oo 

Wilson's  Treatise  on  Steam-boilers.     (Flather.) i6mo,  2  50 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  . .  .8vo,  4  oo 


MECHANICS   AND  MACHINERY. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Chase's  The  Art  of  Pattern-making i2mo,  2  50 

Chordal. — Extracts  from  Letters i2mo,  2  oo 

Church's  Mechanics  of  Engineering 8vo>  6  oo 

Notes  and  Examples  in  Mechanics 8vo,  2  oo 

Compton's  First  Lessons  in  Metal- working I2mo,  i  50 

Compton  and  De  Groodt's  The  Speed  Lathe i2mo,  i  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i  50 

Treatise  on  Belts  and  Pulleys I2mo,  i  50 

Dana's  Text-book  of  Elementary  Mechanics  for  the  Use  of  Colleges  and 

Schools i2mo,  i  50 

Dingey's  Machinery  Pattern  Making I2mo,  2  oo 

Dredge's  Record   of  the   Transportation   Exhibits   Building  of  the   World's 

Columbian  Exposition  of  1893 4to,  half  morocco,  5  oo 

Du  Bois's  Elementary  Principles  of  Mechanics : 

Vol.     I.— Kinematics 8vo,  3  So 

Vol.    II.— Statics 8vo,  4  oo 

Vol.  III.— Kinetics 8vo,  3  So 

Mechanics  of  Engineering.     Vol.   I Small  4to,  7  BO 

Vol.  II Small  4to,  10  oo 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Fitzgerald's  Boston  Machinist i6mo,  i  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power 12010,  3  oo 

Rope  Driving i2mo,  2  oo 

Goss's  Locomotive  Sparks .  v 8vo,  2  oo 

Hall's  Car  Lubrication i2mo,  i  oo 

Holly's  Art  of  Saw  Filing i8mo  75    * 

*  Johnson's  Theoretical  Mechanics I2mo,  3  oo 

Statics  by  Graphic  and  Algebraic  Methods 8vo,  2  oo 

Jones's  Machine  Design: 

Part   I. — Kinematics  of  Machinery 8vo,  i  50 

Part  H. — Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Lanza's  Applied  Mechanics 8vo,  7  50 

MacCord's  Kinematics;   or,  Practical  Mechanism 8vo,  5  oo 

Velocity  Diagrams 8vo,  i  50 

Maurer's  Technical  Mechanics.     (In  preparation.} 

13 


Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo,  4  oo 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  oo 

Reagan's  Locomotives:  Simple,  Compound,  and  Electric i2mo,  2  50 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design .  .  8vo,  3  oo 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism Svo,  3  oo 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     (In  preparation.} 

Sinclair's  Locomotive-engine  Running  and  Management i2mo,  2  oo 

Smith's  Press-working  of  Metals ,; Svo,  3  oo 

Materials  of  Machines .' i2mo,  i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering Svo,  3  oo 

Thurston's  Treatise  on  Friction  and  Lost  Work  in  Machinery  and   Mill 

Work Svo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics.  i2mo,  i  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing Svo,  7  50 

Weisbach's    Kinematics    and    the   Power  of    Transmission.     (Herrmann — 

Klein.) Svo,  5  oo 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.). Svo,  5  oo 

Wood's  Elements  of  Analytical  Mechanics Svo,  3  oo 

Principles  of  Elementary  Mechanics i2mo,  i  25 

Turbines Svo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

METALLURGY. 

Egleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

Vol.   I. — Silver Svo,  7  So 

Vol.    II. — Gold  and  Mercury Svo,  7  So 

**  Iles's  Lead-smelting.     (Postage  9  cents  additional.) i2mo,  2  50 

Keep's  Cast  Iron ; Svo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe Svo,  i  50 

Le  Chatelier's  High-temperature  Measurements.   (Boudouard — Burgess.) .  i2mo,  3  oo 

Metcalf' s  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Smith's  Materials  of  Machines i2mo,  i  oo 

Thurston's  Materials  of  Engineering.     In  Three  Parts Svo,  8  oo 

Part   II. — Iron  and  Steel Svo,  3  50 

Part  III.— A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and   their 

Constituents Svo,  2~so 

UlkeV  Modern  Electrolytic  Copper  Refining Svo,  f  3  oo 

MINERALOGY. 

Barringer's  Description  of  Minerals  of  Commercial  Value.     Oblong,  morocco,  2  50 

Boyd's  Resources  of  Southwest  Virginia Svo,  3  oo 

Map  of  Southwest  Virginia Pocket-book  form,  2  oo 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) Svo,  4  oo 

Chester's  Catalogue  of  Minerals Svo,  paper,  i  oo 

Cloth,  i  25 

Dictionary  of  the  Names  of  Minerals Svo,  3  50 

Dana's  System  of  Mineralogy Large  Svo,  half  leather,  12  50 

First  Appendix  to  Dana's  New  "System  of  Mineralogy." .  .  .  .Large  Svo,  i  oo 

Text-book  of  Mineralogy Svo,  4  oo 

Minerals  and  How  to  Study  Them i2mo,  i  50 

Catalogue  of  American  Localities  of  Minerals Large  Svo,  i  co 

Manual  of  Mineralogy  and  Petrography I2mo,  2  oo 

Egleston's  Catalogue  of  Minerals  and  Synonyms Svo,  2  50 

Hussak's  The  Determination  of  Rock-forming  Minerals.     (Smith.)  Small  Svo,  2  oo 

14 


*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,  o  50 
Rosenbusch's   Microscopical   Physiography   of   the    Rock-making   Minerals. 

(Iddings.) 8vo,    5  oo 

*  Tillman's  Text-book  of  Important  Minerals  and  Docks 8vo,  2  oo 

Williams's  Manual  of  Lithology 8vo,  3  oo 

MINING. 

Beard's  Ventilation  of  Mines i2mo,  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo,  3  oo 

Map  of  Southwest  Virginia Pocket-book  form,  2  oo 

*  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills. 

4to,  half  morocco,  25  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Fowler's  Sewage  Works  Analyses i2mo,  oo 

Goodyear's  Coal-mines  of  the  Western  Coast  of  the  United  States i2mo,  50 

Ihlseng's  Manual  of  Mining 8vo,  oo 

**  Iles's  Lead-smelting.     (Postage  oc.  additional.) i2mo,  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe. .  .... . . ;-.-,,•  • -,.  -,-  •  •  •  •  •  •  -8vo,  50 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  oo 

*  Walke's  Lectures  on  Explosives 8vo,  oo 

Wilson's  Cyanide  Processes i2mo,  50 

Chlorination  Process i2mo,  50 

Hydraulic  and  Placer  Mining i2mo,  2  oo 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation 12 mo,  i  25 


SANITARY  SCIENCE. 

Copeland's  Manual  of  Bacteriology.     (In  preparation.') 

Folwell's  Sewerage.     (Designing,  Construction,  and  Maintenance.) 8vo,  3  oo 

Water-supply  Engineering 8vo,  4  oo 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration   Works i2mo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  i  oo 

Goodrich's  Economical  Disposal  of  Town's  Refuse Demy  8vo,  3  50 

Hazen's  Filtration  of  Public  Water-supplies 8vo,  3  oo 

Kiersted's  Sewage  Disposal , i2mo,  i  25 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control.     (In  preparation.} 

Mason's    Water-supply.     (Considered    Principally   from    a    Sanitary    Stand- 
point.)    3d  Edition,  Rewritten 8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  i  25 

Merriman's  Elements  of  Sanitary  Engineering 8vo,  2  oo 

Nichols's  Water-supply.     (Considered  Mainly  from  a  Chemical  and  Sanitary 

Standpoint.)     (1883.); 8vo,  2  50 

Ogden's  Sewer  Design , i2mo,  2  oo 

*  Price's  Handbook  on  Sanitation i2mo,  i  50 

Richards's  Cost  of  Food.     A  Study  in  Dietaries i2mo,  i  oo 

Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  i  oo 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Stand- 
point   8vo,  2  oo 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  i  50 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage 8vo,  3  50 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Woodhull's  Notes  andJMilitary^Hygiene i6mo,  i  50 

15 


MISCELLANEOUS. 

Barker's  Deep-sea  Soundings 8vo,  2  oo 

Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  8vo,  i  50 

Ferrel's  Popular  Treatise  on  the  Winds 8vo,  4  oo 

Haines's  American  Railway  Management i2mo,  2  50 

Mott's  Composition,  Digestibility,  and  Nutritive  Value  of  Food.   Mounted  chart,  i  25 

Fallacy  of  the  Present  Theory  of  Sound i6mo,  i  oo 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1824-1894.  Small  8vo,  3  oo 

Rotherham's  Emphasized  New  Testament.  /. Large  8vo,  2  oo 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

Totten's  Important  Question  in  Metrology 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

Worcester  and  Atkinson.     Small  Hospitals,  Establishment  and  Maintenance, 
and  Suggestions  for  Hospital  Architecture,  with  Plans  for  a  Small 

Hospital I2mo,  i  25 

HEBREW  AND  CHALDEE    TEXT-BOOKS. 

Green's  Grammar  of  the  Hebrew  Language 8vo,  3  oo 

Elementary  Hebrew  Grammar i2mo,  i  25 

Hebrew  Chrestomathy 8vo,  2  oo 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco,  5  oo 

Letteris's  Hebrew  Bible 8vo,  2  25 

16 


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SEP  14  1915 
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1918 


MAY  11 1918 


13  W19 


31 

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